2.8: Linear Inequalities and Absolute Value Inequalities
- Page ID
- 114993
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this section, you will:
- Use interval notation
- Use properties of inequalities.
- Solve inequalities in one variable algebraically.
- Solve absolute value inequalities.
It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.
Using Interval Notation
Indicating the solution to an inequality such as can be achieved in several ways.
We can use a number line as shown in Figure 2. The blue ray begins at and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.
We can use set-builder notation: which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.
The third method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to are represented as This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.
The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are or all numbers between and including but not including all real numbers between, but not including and and all real numbers less than and including Table 1 outlines the possibilities.
Set Indicated | Set-Builder Notation | Interval Notation |
---|---|---|
All real numbers between a and b, but not including a or b | ||
All real numbers greater than a, but not including a | ||
All real numbers less than b, but not including b | ||
All real numbers greater than a, including a | ||
All real numbers less than b, including b | ||
All real numbers between a and b, including a | ||
All real numbers between a and b, including b | ||
All real numbers between a and b, including a and b | ||
All real numbers less than a or greater than b | ||
All real numbers |
Using Interval Notation to Express All Real Numbers Greater Than or Equal to a
Use interval notation to indicate all real numbers greater than or equal to
- Answer
Use a bracket on the left of and parentheses after infinity: The bracket indicates that is included in the set with all real numbers greater than to infinity.
Use interval notation to indicate all real numbers between and including and
Using Interval Notation to Express All Real Numbers Less Than or Equal to a or Greater Than or Equal to b
Write the interval expressing all real numbers less than or equal to or greater than or equal to
- Answer
We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at and ends at which is written as
The second interval must show all real numbers greater than or equal to which is written as However, we want to combine these two sets. We accomplish this by inserting the union symbol, between the two intervals.
Express all real numbers less than or greater than or equal to 3 in interval notation.
Using the Properties of Inequalities
When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.
These properties also apply to and
Demonstrating the Addition Property
Illustrate the addition property for inequalities by solving each of the following:
- ⓐ
- ⓑ
- ⓒ
- Answer
The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.
- ⓐ
- ⓑ
- ⓒ
- ⓐ
Solve:
Demonstrating the Multiplication Property
Illustrate the multiplication property for inequalities by solving each of the following:
- ⓐ
- ⓑ
- ⓒ
- Answer
- ⓐ
- ⓑ
- ⓒ
- ⓐ
Solve:
Solving Inequalities in One Variable Algebraically
As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.
Solving an Inequality Algebraically
Solve the inequality:
- Answer
Solving this inequality is similar to solving an equation up until the last step.
The solution set is given by the interval or all real numbers less than and including 1.
Solve the inequality and write the answer using interval notation:
Solving an Inequality with Fractions
Solve the following inequality and write the answer in interval notation:
- Answer
We begin solving in the same way we do when solving an equation.
The solution set is the interval
Solve the inequality and write the answer in interval notation:
Understanding Compound Inequalities
A compound inequality includes two inequalities in one statement. A statement such as means and There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.
Solving a Compound Inequality
Solve the compound inequality:
- Answer
The first method is to write two separate inequalities: and We solve them independently.
Then, we can rewrite the solution as a compound inequality, the same way the problem began.
In interval notation, the solution is written as
The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.
We get the same solution:
Solve the compound inequality:
Solving a Compound Inequality with the Variable in All Three Parts
Solve the compound inequality with variables in all three parts:
- Answer
Let's try the first method. Write two inequalities:
The solution set is or in interval notation Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line. See Figure 3.
Figure 3
Solve the compound inequality:
Solving Absolute Value Inequalities
As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at has an absolute value of as it is x units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.
An absolute value inequality is an equation of the form
Where A, and sometimes B, represents an algebraic expression dependent on a variable x. Solving the inequality means finding the set of all -values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of x-values such that the distance between and 600 is less than or equal to 200. We represent the distance between and 600 as and therefore, or
This means our returns would be between $400 and $800.
To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.
For an algebraic expression X, and an absolute value inequality is an inequality of the form
These statements also apply to and
Determining a Number within a Prescribed Distance
Describe all values within a distance of 4 from the number 5.
- Answer
We want the distance between and 5 to be less than or equal to 4. We can draw a number line, such as in Figure 4, to represent the condition to be satisfied.
Figure 4
The distance from to 5 can be represented using an absolute value symbol, Write the values of that satisfy the condition as an absolute value inequality.
We need to write two inequalities as there are always two solutions to an absolute value equation.
If the solution set is and then the solution set is an interval including all real numbers between and including 1 and 9.
So is equivalent to in interval notation.
Describe all x-values within a distance of 3 from the number 2.
Solving an Absolute Value Inequality
Solve .
- Answer
Using a Graphical Approach to Solve Absolute Value Inequalities
Given the equation determine the x-values for which the y-values are negative.
- Answer
We are trying to determine where which is when We begin by isolating the absolute value.
Next, we solve for the equality
Now, we can examine the graph to observe where the y-values are negative. We observe where the branches are below the x-axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at and and that the graph opens downward. See Figure 5.
Figure 5
Solve
Access these online resources for additional instruction and practice with linear inequalities and absolute value inequalities.
2.7 Section Exercises
Verbal
When solving an inequality, explain what happened from Step 1 to Step 2:
When solving an inequality, we arrive at:
Explain what our solution set is.
When writing our solution in interval notation, how do we represent all the real numbers?
When solving an inequality, we arrive at:
Explain what our solution set is.
Describe how to graph
Algebraic
For the following exercises, solve the inequality. Write your final answer in interval notation.
For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.
For the following exercises, describe all the x-values within or including a distance of the given values.
Distance of 5 units from the number 7
Distance of 3 units from the number 9
Distance of 10 units from the number 4
Distance of 11 units from the number 1
For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.
Graphical
For the following exercises, graph the function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.
For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines.
Numeric
For the following exercises, write the set in interval notation.
For the following exercises, write the interval in set-builder notation.
For the following exercises, write the set of numbers represented on the number line in interval notation.
Technology
For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2nd CALC 5:intersection, 1st curve, enter, 2nd curve, enter, guess, enter). Copy a sketch of the graph and shade the x-axis for your solution set to the inequality. Write final answers in interval notation.
Extensions
Solve
Solve
is a profit formula for a small business. Find the set of x-values that will keep this profit positive.
Real-World Applications
In chemistry the volume for a certain gas is given by where V is measured in cc and T is temperature in ºC. If the temperature varies between 80ºC and 120ºC, find the set of volume values.
A basic cellular package costs $20/mo. for 60 min of calling, with an additional charge of $.30/min beyond that time.. The cost formula would be If you have to keep your bill no greater than $50, what is the maximum calling minutes you can use?