Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

2.7: Solve Linear Inequalities

  • Page ID
    18938
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Learning Objectives

    By the end of this section, you will be able to:

    • Graph inequalities on the number line
    • Solve inequalities using the Subtraction and Addition Properties of inequality
    • Solve inequalities using the Division and Multiplication Properties of inequality
    • Solve inequalities that require simplification
    • Translate to an inequality and solve

    Note

    Before you get started, take this readiness quiz.

    1. Translate from algebra to English: \(15>x\).
      If you missed this problem, review [link].
    2. Solve: \(n−9=−42\).
      If you missed this problem, review [link].
    3. Solve: \(−5p=−23\).
      If you missed this problem, review [link].
    4. Solve: \(3a−12=7a−20\).
      If you missed this problem, review [link].

    Graph Inequalities on the Number Line

    Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

    What about the solution of an inequality? What number would make the inequality \(x > 3\) true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality \(x > 3\).

    We show the solutions to the inequality \(x > 3\) on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of \(x > 3\) is shown in Figure \(\PageIndex{1}\). Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.

    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis.
    Figure \(\PageIndex{1}\): The inequality \(x > 3\) is graphed on this number line.

    The graph of the inequality \(x \geq 3\) is very much like the graph of \(x > 3\), but now we need to show that 3 is a solution, too. We do that by putting a bracket at \(x = 3\), as shown in Figure \(\PageIndex{2}\).

    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on the number line, with an open bracket at x equals 3, and a red line extending to the right of the bracket.
    Figure \(\PageIndex{2}\): The inequality \(x \geq 3\) is graphed on this number line.

    Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

    Exercise \(\PageIndex{1}\)

    Graph on the number line:

    1. \(x\leq 1\)
    2. \(x<5\)
    3. \(x>−1\)
    Answer

    1. \(x\leq 1\) This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at x=1 to show that it is included.
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket.

    2. \(x<5\) This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at x=5 to show it is not included.
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 5 is graphed on the number line, with an open parenthesis at x equals 5, and a red line extending to the right of the parenthesis.

    3. \(x>−1\) This means all numbers greater than −1, but not including −1. We shade in all the numbers on the number line to the right of −1, then put a parenthesis at x=−1 to show it is not included.
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 1 is graphed on the number line, with an open parenthesis at x equals negative 1, and a red line extending to the right of the parenthesis.

    Exercise \(\PageIndex{2}\)

    Graph on the number line:

    1. \(x\leq −1\)
    2. \(x>2\)
    3. \(x<3\)
    Answer

     

    1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the left of the bracket.
    2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis.
    3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the left of the parenthesis.

    Exercise \(\PageIndex{3}\)

    Graph on the number line:

    1. \(x>−2\)
    2. \(x<−3\)
    3. \(x\geq −1\)
    Answer

     

    1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 2 is graphed on the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the right of the parenthesis.
    2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.
    3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket.

    We can also represent inequalities using interval notation. As we saw above, the inequality \(x>3\) means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express \(x>3\) as \((3, \infty)\). The symbol \(\infty\) is read as ‘infinity’. It is not an actual number. Figure \(\PageIndex{3}\) shows both the number line and the interval notation.

    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 3 comma infinity, parenthesis.
    Figure \(\PageIndex{3}\): The inequality \(x>3\) is graphed on this number line and written in interval notation.

    The inequality \(x\leq 1\) means all numbers less than or equal to 1. There is no lower end to those numbers. We write \(x\leq 1\) in interval notation as \((-\infty, 1]\). The symbol \(-\infty\) is read as ‘negative infinity’. Figure \(\PageIndex{4}\) shows both the number line and interval notation.

    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket.
    Figure \(\PageIndex{4}\): The inequality \(x\leq 1\) is graphed on this number line and written in interval notation.

    INEQUALITIES, NUMBER LINES, AND INTERVAL NOTATION

    This figure show four number lines, all without tick marks. The inequality x is greater than a is graphed on the first number line, with an open parenthesis at x equals a, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, a comma infinity, parenthesis. The inequality x is greater than or equal to a is graphed on the second number line, with an open bracket at x equals a, and a red line extending to the right of the bracket. The inequality is also written in interval notation as bracket, a comma infinity, parenthesis. The inequality x is less than a is graphed on the third number line, with an open parenthesis at x equals a, and a red line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma a, parenthesis. The inequality x is less than or equal to a is graphed on the last number line, with an open bracket at x equals a, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma a, bracket.

    Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure \(\PageIndex{5}\).

    This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.”
    Figure \(\PageIndex{5}\): The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals.

    Exercise \(\PageIndex{4}\)

    Graph on the number line and write in interval notation.

    1. \(x \geq -3\)
    2. \(x<2.5\)
    3. \(x\leq \frac{3}{5}\)
    Answer

    1.

      .
    Shade to the right of −3, and put a bracket at −3. .
    Write in interval notation. .
    2.
      .
    Shade to the left of 2.5, and put a parenthesis at 2.5. .
    Write in interval notation. .
    3.
      .
    Shade to the left of \(-\frac{3}{5}\), and put a bracket at \(-\frac{3}{5}\). .
    Write in interval notation. .

    Exercise \(\PageIndex{5}\)

    Graph on the number line and write in interval notation:

    1. \(x>2\)
    2. \(x\leq −1.5\)
    3. \(x\geq \frac{3}{4}\)
    Answer

     

    1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 2 comma infinity, parenthesis.
    2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 1.5, bracket.
    3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3/4 is graphed on the number line, with an open bracket at x equals 3/4, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 3/4 comma infinity, parenthesis.

    Exercise \(\PageIndex{6}\)

    Graph on the number line and write in interval notation:

    1. \(x\leq −4\)
    2. \(x\geq 0.5\)
    3. \(x<-\frac{2}{3}\)
    Answer

     

    1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 4 is graphed on the number line, with an open bracket at x equals negative 4, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 4, bracket.
    2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 0.5 is graphed on the number line, with an open bracket at x equals 0.5, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, o.5 comma infinity, parenthesis.
    3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 2/3 is graphed on the number line, with an open parenthesis at x equals negative 2/3, and a dark line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma negative 2/3, parenthesis.

    Solve Inequalities using the Subtraction and Addition Properties of Inequality

    The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

    PROPERTIES OF EQUALITY

    \[\begin{array} { l l } { \textbf { Subtraction Property of Equality } } & { \textbf { Addition Property of Equality } } \\ { \text { For any numbers } a , b , \text { and } c , } & { \text { For any numbers } a , b , \text { and } c } \\ { \text { if } \qquad \quad a = b , } & { \text { if } \qquad \quad a = b } \\ { \text { then } a - c = b - c . } & { \text { then } a + c = b + c } \end{array}\]

    Similar properties hold true for inequalities.

    For example, we know that −4 is less than 2. .
    If we subtract 5 from both quantities, is the left side still less than the right side? .
    We get −9 on the left and −3 on the right. .
    And we know −9 is less than −3. .
     

    The inequality sign stayed the same.

    Table \(\PageIndex{1}\)

    Similarly we could show that the inequality also stays the same for addition.

    This leads us to the Subtraction and Addition Properties of Inequality.

    PROPERTIES OF INEQUALITY

    \[\begin{array} { l l } { \textbf { Subtraction Property of Inequality } } & { \textbf { Addition Property of Inequality } } \\ { \text { For any numbers } a , b , \text { and } c , } & { \text { For any numbers } a , b , \text { and } c } \\ { \text { if }\qquad \quad a < b } & { \text { if } \qquad \quad a < b } \\ { \text { then } a - c < b - c . } & { \text { then } a + c < b + c } \\\\ { \text { if } \qquad \quad a > b } & { \text { if } \qquad \quad a > b } \\ { \text { then } a - c > b - c . } & { \text { then } a + c > b + c } \end{array}\]

    We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality \(x+5>9\), the steps would look like this:

    \[\begin{array}{rrll} {} &{x + 5} &{ >} &{9} \\ {\text{Subtract 5 from both sides to isolate }x.} &{x + 5 - 5} &{ >} &{9 - 5} \\{} &{x} &{ >} &{4} \\ \end{array}\]

    Any number greater than 4 is a solution to this inequality.

    Exercise \(\PageIndex{7}\)

    Solve the inequality \(n - \frac{1}{2} \leq \frac{5}{8}\), graph the solution on the number line, and write the solution in interval notation.

    Answer
      .
    Add \(\frac{1}{2}\) to both sides of the inequality. .
    Simplify. .
    Graph the solution on the number line. .
    Write the solution in interval notation.  

    Exercise \(\PageIndex{8}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(p - \frac{3}{4} \geq \frac{1}{6}\)

    Answer

    This figure shows the inequality p is greater than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at p equals 11/12, and a dark line extends to the right from 11/12. Below the number line is the solution written in interval notation: bracket, 11/12 comma infinity, parenthesis.

    Exercise \(\PageIndex{9}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(r - \frac{1}{3} \leq \frac{7}{12}\)

    Answer

    This figure shows the inequality r is less than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at r equals 11/12, and a dark line extends to the left from 11/12. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/12, bracket.

    Solve Inequalities using the Division and Multiplication Properties of Inequality

    The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

    PROPERTIES OF EQUALITY

    \[\begin{array}{ll} {\textbf{Division Property of Equality}} &{\textbf{MUltiplication Property of Equality}} \\ {\text{For any numbers a, b, c, and c} \neq 0} &{\text{For any numbers a, b, c}} \\ {\text{if } \qquad a = b} &{\text{if} \qquad \quad a = b} \\ {\text{then }\quad \frac{a}{c} = \frac{b}{c}} &{\text{then } \quad ac = bc} \end{array}\]

    Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?

    Consider some numerical examples.

      .   .
    Divide both sides by 5. . Multiply both sides by 5. .
    Simplify. .   .
    Fill in the inequality signs. .   .

    Table \(\PageIndex{2}\)

    The inequality signs stayed the same.

    Does the inequality stay the same when we divide or multiply by a negative number?

      .   .
    Divide both sides by -5. . Multiply both sides by -5. .
    Simplify. .   .
    Fill in the inequality signs. .   .

    Table \(\PageIndex{3}\)

    The inequality signs reversed their direction.

    When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

    Here are the Division and Multiplication Properties of Inequality for easy reference.

    DIVISION AND MULTIPLICATION PROPERTIES OF INEQUALITY

    For any real numbers a,b,c

    \[\begin{array}{ll} {\text{if } a < b \text{ and } c > 0, \text{ then}} &{\frac{a}{c} < \frac{b}{c} \text{ and } ac < bc} \\ {\text{if } a > b \text{ and } c > 0, \text{ then}} &{\frac{a}{c} > \frac{b}{c} \text{ and } ac > bc} \\ {\text{if } a < b \text{ and } c < 0, \text{ then}} &{\frac{a}{c} > \frac{b}{c} \text{ and } ac > bc} \\ {\text{if } a > b \text{ and } c < 0, \text{ then}} &{\frac{a}{c} < \frac{b}{c} \text{ and } ac < bc} \end{array}\]

    When we divide or multiply an inequality by a:

    • positive number, the inequality stays the same.
    • negative number, the inequality reverses.

    Exercise \(\PageIndex{10}\)

    Solve the inequality \(7y<​​42\), graph the solution on the number line, and write the solution in interval notation.

    Answer
      .
    Divide both sides of the inequality by 7.
    Since \(7>0\), the inequality stays the same.
    .
    Simplify. .
    Graph the solution on the number line. .
    Write the solution in interval notation. .

    Exercise \(\PageIndex{11}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(9c>72\)

    Answer

    \(c>8\)

    This figure is a number line ranging from 6 to 10 with tick marks for each integer. The inequality c is greater than 8 is graphed on the number line, with an open parenthesis at c equals 8, and a dark line extending to the right of the parenthesis.

    Exercise \(\PageIndex{12}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(12d\leq 60\)

    Answer

    \(d\leq 5\)

    This figure is a number line ranging from 3 to 7 with tick marks for each integer. The inequality d is less than or equal to 5 is graphed on the number line, with an open bracket at d equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.

    Exercise \(\PageIndex{13}\)

    Solve the inequality \(−10a\geq 50\), graph the solution on the number line, and write the solution in interval notation.

    Answer
      .
    Divide both sides of the inequality by −10.
    Since \(−10<0\), the inequality reverses.
    .
    Simplify. .
    Graph the solution on the number line. .
    Write the solution in interval notation. .

    Exercise \(\PageIndex{14}\)

    Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    \(−8q<32\)

    Answer

    \(q>−4\)

    This figure is a number line ranging from negative 6 to negative 3 with tick marks for each integer. The inequality q is greater than negative 4 is graphed on the number line, with an open parenthesis at q equals negative 4, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 4 comma infinity, parenthesis.

    Exercise \(\PageIndex{15}\)

    Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    \(−7r\leq −70\)

    Answer

    This figure is a number line ranging from 9 to 13 with tick marks for each integer. The inequality r is greater than or equal to 10 is graphed on the number line, with an open bracket at r equals 10, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 10 comma infinity, parenthesis.

    SOLVING INEQUALITIES

    Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.

    \[\begin{array}{l} x > a\text{ has the same meaning as } a < x \end{array}\]

    Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”

    Exercise \(\PageIndex{16}\)

    Solve the inequality \(-20 < \frac{4}{5}u\), graph the solution on the number line, and write the solution in interval notation.

    Answer
      .
    Multiply both sides of the inequality by \(\frac{5}{4}\).
    Since \(\frac{5}{4} > 0\), the inequality stays the same.
    .
    Simplify. .
    Rewrite the variable on the left. .
    Graph the solution on the number line. .
    Write the solution in interval notation. .

    Exercise \(\PageIndex{17}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(24 \leq \frac{3}{8}m\)

    Answer

    This figure shows the inequality m is greater than or equal to 64. Below this inequality is a number line ranging from 63 to 67 with tick marks for each integer. The inequality m is greater than or equal to 64 is graphed on the number line, with an open bracket at m equals 64, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 64 comma infinity, parenthesis.

    Exercise \(\PageIndex{18}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(-24 < \frac{4}{3}n\)

    Answer

    This figure shows the inequality n is greater than negative 18. Below this inequality is a number line ranging from negative 20 to negative 16 with tick marks for each integer. The inequality n is greater than negative 18 is graphed on the number line, with an open parenthesis at n equals negative 18, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 18 comma infinity, parenthesis.

    Exercise \(\PageIndex{19}\)

    Solve the inequality \(\frac{t}{-2} \geq 8\), graph the solution on the number line, and write the solution in interval notation.

    Answer
      .
    Multiply both sides of the inequality by −2.
    Since \(−2<0\), the inequality reverses.
    .
    Simplify. .
    Graph the solution on the number line. .
    Write the solution in interval notation. .

    Exercise \(\PageIndex{20}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(\frac{k}{-12}\leq 15\)

    Answer

    This figure shows the inequality k is greater than or equal to negative 180. Below this inequality is a number line ranging from negative 181 to negative 177 with tick marks for each integer. The inequality k is greater than or equal to negative 180 is graphed on the number line, with an open bracket at n equals negative 180, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, negative 180 comma infinity, parenthesis.

    Exercise \(\PageIndex{21}\)

    Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

    \(\frac{u}{-4}\geq -16\)

    Answer

    This figure shows the inequality u is less than or equal to 64. Below this inequality is a number line ranging from 62 to 66 with tick marks for each integer. The inequality u is less than or equal to 64 is graphed on the number line, with an open bracket at u equals 64, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 64, bracket.

    ​​​​​

    Solve Inequalities That Require Simplification

    Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.

    Exercise \(\PageIndex{22}\)

    Solve the inequality \(4m\leq 9m+17\), graph the solution on the number line, and write the solution in interval notation.

    Answer
      .
    Subtract 9m from both sides to collect the variables on the left. .
    Simplify. .
    Divide both sides of the inequality by −5, and reverse the inequality. .
    Simplify. .
    Graph the solution on the number line. .
    Write the solution in interval notation. .

    Exercise \(\PageIndex{23}\)

    Solve the inequality \(3q\geq 7q−23\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows the inequality q is less than or equal to 23/4. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality q is less than or equal to 23/4 is graphed on the number line, with an open bracket at q equals 23/4 (written in), and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 23/4, bracket.

    Exercise \(\PageIndex{24}\)

    Solve the inequality \(6x<10x+19\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows the inequality x is greater than negative 19/4. Below this inequality is a number line ranging from negative 7 to negative 3, with tick marks for each integer. The inequality x is greater than negative 19/4 is graphed on the number line, with an open parenthesis at x equals negative 19/4 (written in), and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 19/4 comma infinity, parenthesis.

    Exercise \(\PageIndex{25}\)

    Solve the inequality \(8p+3(p−12)>7p−28\) graph the solution on the number line, and write the solution in interval notation.

    Answer
    Simplify each side as much as possible. 8p+3(p−12)>7p−28
    Distribute. 8p+3p−36>7p−28
    Combine like terms. 11p−36>7p−28
    Subtract 7p from both sides to collect the variables on the left. 11p−36−7p>7p−28−7p
    Simplify. 4p−36>−28
    Add 36 to both sides to collect the constants on the right. 4p−36+36>−28+36
    Simplify. 4p>8
    Divide both sides of the inequality by 4; the inequality stays the same. \(\frac{4p}{4}>84\)
    Simplify. \(p>2\)
    Graph the solution on the number line. .
    Write the solution in interval notation. \((2, \infty)\)

    Exercise \(\PageIndex{26}\)

    Solve the inequality \(9y+2(y+6)>5y−24\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows the inequality y is greater than negative 6. Below this inequality is a number line ranging from negative 7 to negative 3 with tick marks for each integer. The inequality y is greater than negative 6 is graphed on the number line, with an open parenthesis at y equals negative 6, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 6 comma infinity, parenthesis.

    Exercise \(\PageIndex{27}\)

    Solve the inequality \(6u+8(u−1)>10u+32\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows the inequality u is greater than 10. Below this inequality is a number line ranging from 9 to 13 with tick marks for each integer. The inequality u is greater than 10 is graphed on the number line, with an open parenthesis at u equals 10, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 10 comma infinity, parenthesis.

    Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

    Exercise \(\PageIndex{28}\)

    Solve the inequality \(8x−2(5−x)<4(x+9)+6x\), graph the solution on the number line, and write the solution in interval notation.

    Answer
    Simplify each side as much as possible. 8x−2(5−x)<4(x+9)+6x
    Distribute. 8x−10+2x<4x+36+6x
    Combine like terms. 10x−10<10x+36
    Subtract 10x from both sides to collect the variables on the left. 10x−10−10x<10x+36−10x
    Simplify. −10<36
    The xx’s are gone, and we have a true statement. The inequality is an identity.
    The solution is all real numbers.
    Graph the solution on the number line. .
    Write the solution in interval notation. \((-\infty, \infty)\)

    Exercise \(\PageIndex{29}\)

    Solve the inequality \(4b−3(3−b)>5(b−6)+2b\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.

    Exercise \(\PageIndex{30}\)

    Solve the inequality \(9h−7(2−h)<8(h+11)+8h\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.

    Exercise \(\PageIndex{31}\)

    Solve the inequality \(\frac{1}{3}a - \frac{1}{8}a > \frac{5}{24}a + \frac{3}{4}\), graph the solution on the number line, and write the solution in interval notation.

    Answer
      .
    Multiply both sides by the LCD, 24, to clear the fractions. .
    Simplify. .
    Combine like terms. .
    Subtract 5a from both sides to collect the variables on the left. .
    Simplify. .
    The statement is false! The inequality is a contradiction.
      There is no solution.
    Graph the solution on the number line. .
    Write the solution in interval notation. There is no solution.

    Exercise \(\PageIndex{32}\)

    Solve the inequality \(\frac{1}{4}x - \frac{1}{12}x > \frac{1}{6}x + \frac{7}{8}\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”

    Exercise \(\PageIndex{33}\)

    Solve the inequality \(\frac{2}{5}z - \frac{1}{3}z < \frac{1}{15}z - \frac{3}{5}\), graph the solution on the number line, and write the solution in interval notation.

    Answer

    This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement: “No solution.”

    Translate to an Inequality and Solve

    To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.

    Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.

    Table \(\PageIndex{4}\) shows some common phrases that indicate inequalities.

    > \(\geq\) < \(\leq\)
    is greater than is greater than or equal to is less than is less than or equal to
    is more than is at least is smaller than is at most
    is larger than is no less than has fewer than is no more than
    exceeds is the minimum is lower than is the maximum

    Table \(\PageIndex{4}\)

    Exercise \(\PageIndex{34}\)

    Translate and solve. Then write the solution in interval notation and graph on the number line.

    Twelve times c is no more than 96.

    Answer
    Translate. .
    Solve—divide both sides by 12. .
    Simplify. .
    Write in interval notation. .
    Graph on the number line. .

    Exercise \(\PageIndex{35}\)

    Translate and solve. Then write the solution in interval notation and graph on the number line.

    Twenty times y is at most 100

    Answer

    This figure shows the inequality 20y is less than or equal to 100, and then its solution: y is less than or equal to 5. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality y is less than or equal to 5 is graphed on the number line, with an open bracket at y equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.

    Exercise \(\PageIndex{36}\)

    Translate and solve. Then write the solution in interval notation and graph on the number line.

    Nine times z is no less than 135

    Answer

    This figure shows the inequality 9z is greater than or equal to 135, and then its solution: z is greater than or equal to 15. Below this inequality is a number line ranging from 14 to 18 with tick marks for each integer. The inequality z is greater than or equal to 15 is graphed on the number line, with an open bracket at z equals 15, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 15 comma infinity, parenthesis.

    Exercise \(\PageIndex{37}\)

    Translate and solve. Then write the solution in interval notation and graph on the number line.

    Thirty less than x is at least 45.

    Answer
    Translate. .
    Solve—add 30 to both sides. .
    Simplify. .
    Write in interval notation. .
    Graph on the number line. .

    Exercise \(\PageIndex{38}\)

    Translate and solve. Then write the solution in interval notation and graph on the number line.

    Nineteen less than p is no less than 47

    Answer

    This figure shows the inequality p minus 19 is greater than or equal to 47, and then its solution: p is greater than or equal to 66. Below this inequality is a number line ranging from 65 to 69 with tick marks for each integer. The inequality p is greater than or equal to 66 is graphed on the number line, with an open bracket at p equals 66, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 66 comma infinity, parenthesis.

    Exercise \(\PageIndex{39}\)

    Translate and solve. Then write the solution in interval notation and graph on the number line.

    Four more than a is at most 15.

    Answer

    This figure shows the inequality a plus 4 is less than or equal to 15, and then its solution: a is less than or equal to 11. Below this inequality is a number line ranging from 10 to 14 with tick marks for each integer. The inequality a is less than or equal to 11 is graphed on the number line, with an open bracket at a equals 11, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity 11, bracket.

    Key Concepts

    • Subtraction Property of Inequality
      For any numbers a, b, and c,
      if a<b then a−c<b−c and
      if a>b then a−c>b−c.
    • Addition Property of Inequality
      For any numbers a, b, and c,
      if a<b then a+c<b+c and
      if a>b then a+c>b+c.
    • Division and Multiplication Properties of Inequality
      For any numbers a, b, and c,
      if a<b and c>0, then ac<bc and ac>bc.
      if a>b and c>0, then ac>bc and ac>bc.
      if a<b and c<0, then ac>bc and ac>bc.
      if a>b and c<0, then ac<bc and ac<bc.
    • When we divide or multiply an inequality by a:
      • positive number, the inequality stays the same.
      • negative number, the inequality reverses.