# 3.1: Linear Inequalities

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When there is a solution to an equation such as $$x = 4$$, this solution is unique and is the only solution that makes the statement true. However, with inequalities, the solution is an interval of numbers in which make the inequality true.

##### Definition: Inequality

An inequality is a relation between two numbers or two sets of numbers (or elements) in which are not strictly equal, but greater than (or equal to) or less than (or equal to).

$\begin{array}{ll} >&\text{greater than} \\ \geq&\text{greater than or equal to} \\ <&\text{less than} \\ \leq&\text{less than or equal to}\end{array}\nonumber$

##### Note

English mathematician Thomas Harriot first used the inequality symbols in 1631. However, they were not immediately accepted, where symbols $$\sqsubset$$ and $$\sqsupset$$ were already coined by another English mathematician, William Oughtred.

## Graphing Linear Inequalities

If we have an statement such as $$x < 4$$, this means a solution can be any number smaller than $$4$$ such as $$−2,\: 0,\: 3,\: 3.9$$ or even $$3.999999999$$ as long as it is smaller than $$4$$. If we have a statement such as $$x\geq −2$$, this means a solution can be any number greater than or equal to $$−2$$, such as $$5,\: 0,\: −1,\: −1.9999,$$ or even $$−2$$. Because we don’t have one value as the solution, it is often useful to draw a picture of the solutions to the inequality on a number line.

##### Definition: Linear Inequality

A linear inequality in one variable is an inequality of the form $ax + b < c,\nonumber$ where the inequality is written in the same form for $$>,\: ≤,\: ≥$$.

##### Interval Notation

We rewrite the $$>,\: <,\: \leq ,\: \geq$$ symbols as parenthesis and brackets, i.e., $$(,\: ),\: ],\: [,$$ respectively, when we write the inequality in interval notation.

• Case 1. If $$x< a$$, then the equivalent set of numbers in interval notation is $$(-\infty, a)$$; similarly, If $$x\leq a$$, then the equivalent set of numbers in interval notation is $$(-\infty ,a]$$.
• Case 2. If $$x>a$$, then the equivalent set of numbers in interval notation is $$(a, ∞)$$; similarly, If $$x ≥ a$$, then the equivalent set of numbers in interval notation is $$[a, ∞)$$.
##### Example $$\PageIndex{1}$$

Graph the inequality and rewrite the inequality in interval notation: $$x < 2$$

Solution

We will complete this example in steps and use this method for the remaining future examples involving inequalities.

Step 1. Draw a number line and mark the number in the inequality on the line.

Step 2. Write a parenthesis or bracket, depending on the inequality sign, over the number on the number line. Note, we can easily use $$•$$ for $$≤$$ and $$≥$$, and $$◦$$ for $$<$$ and $$>$$. Since we have $$x < 2$$, then we use $$◦$$ or ) on the number line:

Step 3. Draw a line connecting the $$]$$ or $$◦$$ in the direction where lies the solution. Since $$x < 2$$, then we want all numbers less than $$2$$; hence, we draw the line to the left to represent all numbers less than $$2$$:

##### Note

Even though there are two graphs, one will suffice. At the discretion of the instructor, brackets and parenthesis, or closed and open circles will be used when graphing inequalities.

Step 4. Rewrite the inequality in interval notation. Since $$x < 2$$, we can see this is case 1 and so the interval is $$(−∞, 2)$$. We use a parenthesis on the $$2$$ since it is $$<$$.

##### Note

In Example $$\PageIndex{1}$$, we use a parenthesis on $$−∞$$ because $$±∞$$ are not real numbers and symbolizes “some large (positive or negative) number beyond any real number.” It is common practice to always use parentheses on infinities for intervals.

##### Note

The symbol for infinity was first used by the Romans even though, at the time, the number was used for $$1,000$$. The Greeks also used the symbol for $$10,000$$.

##### Example $$\PageIndex{2}$$

Graph the inequality and rewrite the inequality in interval notation: $$y ≥ −1$$

Solution

We start by labeling the number line with $$−1$$. Then draw a line to the right since all numbers greater than (or equal to) $$−1$$ are to the right:

Next, we write $$y ≥ −1$$ in interval notation. This is case 2 and the interval is $$[−1, ∞)$$; we use a bracket because of $$≥$$.

##### Example $$\PageIndex{3}$$

Given the graph, write the equivalent inequality and interval notation:

Solution

Since the graph shows all numbers (strictly) greater than $$3$$, then the inequality is $$x > 3$$; we use $$>$$ because of the $$($$ on the number line. The equivalent interval would be, using case 2, $$(3, ∞)$$.

##### Example $$\PageIndex{4}$$

Given the graph, write the equivalent inequality and interval notation:

Solution

Since the graph shows all numbers less than (or equal to) $$4$$, then the inequality is $$x ≤ 4$$; we use $$≤$$ because of the $$]$$ on the number line. The equivalent interval would be, using case 1, $$(−∞, 4]$$.

## Solving Linear Inequalities

##### Properties of Inequalities

There are two properties are used with inequalities:

Property 1. Addition Property of Inequalities: For numbers $$a,\: b,\: c,$$ $a < b\text{ is equivalent to }a + c < b + c,\nonumber$ where the form is the same for $$>,\: ≤,\: ≥$$. I.e., you can add a number to one side of the inequality as long as you add the same number to the other side. (This should remind you of the addition property for equations.)

Property 2. Multiplication Property of Inequalities: For numbers $$a,\: b,\: c$$ and $$c\neq 0$$, $a < b\text{ is equivalent to }a · c < b · c,\nonumber$ where the form is the same for $$>,\: ≤,\: ≥$$. I.e., you can multiply a nonzero number to one side of the inequality as long as you multiply the same nonzero number to the other side.
However, if $$c < 0$$, then the inequality reverses, i.e, if you multiply or divide by a negative, then reverse the inequality symbol.

##### Example $$\PageIndex{5}$$

Solve the inequality. Graph the solution and write the solution in interval notation.

$5-2x\geq 11\nonumber$

Solution

$\begin{array}{rl}5-2x\geq 11&\text{Add the opposite of }5\text{ to each side} \\ 5-2x+\color{blue}{(-5)}\color{black}{}\geq 11+\color{blue}{(-5)}&\text{Simplify} \\ -2x\geq 6&\text{Multiply by the reciprocal of }-2 \\ \color{blue}{\left(-\frac{1}{2}\right)}\color{black}{}\cdot -2x\leq 6\cdot\color{blue}{\left(-\frac{1}{2}\right)}\color{black}{}&\text{Simplify and reverse the inequality sign} \\ x\color{blue}{\leq}\color{black}{}-3&\text{Solution}\end{array}\nonumber$

Let’s graph the solution $$x ≤ −3$$:

Next, we write $$x ≤ −3$$ in interval notation. This is case 1 and the interval is $$(−∞, −3]$$; we use a bracket because of $$≤$$.

##### Example $$\PageIndex{6}$$

Solve the inequality. Graph the solution and write the solution in interval notation.

$3(2x-4)+4x<4(3x-7)+8\nonumber$

Solution

$\begin{array}{rl}3(2x-4)+4x<4(3x-7)+8&\text{Distribute} \\ 6x-12+4x<12x-28+8&\text{Combine like terms} \\ 10x-12<12x-20&\text{Isolate the variable term} \\ 10x-12+\color{blue}{(-10x)}\color{black}{}<12x-20+\color{blue}{(-10x)}\color{black}{}&\text{Simplify} \\ -12<2x-20&\text{Add the opposite of }-20\text{ to each side} \\ -12+\color{blue}{20}\color{black}{}<2x-20+\color{blue}{20}\color{black}{}&\text{Simplify} \\ 8<2x&\text{Multiply by the reciprocal of }2 \\ \color{blue}{\frac{1}{2}}\color{black}{}\cdot 8<2x\cdot\color{blue}{\frac{1}{2}}&\text{Simplify} \\ 4<x&\text{Rewrite the inequality with }x\text{ on the left side} \\ x>4&\text{Solution}\end{array}\nonumber$

Let's graph the solution $$x>4$$:

Next, we write $$x > 4$$ in interval notation. This is case 2 and the interval is $$(4, ∞)$$; we use a parenthesis because of $$>$$.

##### Note

It is important to be careful when the solution to the inequality has the isolated variable on the right side like in Example $$\PageIndex{6}$$, i.e., $$4 < x$$ rather than $$x > 4$$. It is best practice to write the variable on the left side after isolating the variable. This will minimize confusion when graphing the solution and writing the solution in interval notation.

## Tripartite Inequalities

A special type of compound inequality is called a tripartite inequality, when the variable (or expression containing the variable) is between two numbers. When solving these types of inequalities with three parts, we will apply the properties of inequalities to all three parts (rather than two sides) to isolate the variable in the middle.

##### Interval Notation for Tripartite Inequalities

Case 1. If $$a < x < b$$, then the equivalent set of numbers in interval notation is $$(a, b)$$; similarly, If $$a ≤ x ≤ b$$, then the equivalent set of numbers in interval notation is $$[a, b]$$.

Case 2. If $$a < x ≤ b$$, then the equivalent set of numbers in interval notation is $$(a, b]$$; similarly, If $$a ≤ x < b$$, then the equivalent set of numbers in interval notation is $$[a, b)$$.

##### Example $$\PageIndex{7}$$

Solve the inequality. Graph the solution and write the solution in interval notation.

$-6\leq -4x+2<2\nonumber$

Solution

Solve the inequality, graph the solution, and give interval notation.

$\begin{array}{rl}-6\leq -4x+2<2&\text{Add the opposite of }2\text{ to each part of the inequality} \\ -6+\color{blue}{(-2)}\color{black}{}\leq -4x+2+\color{blue}{(-2)}\color{black}{}<2+\color{blue}{(-2)}\color{black}{}&\text{Simplify} \\ -8\leq -4x<0&\text{Multiply by the reciprocal of }-4 \\ \color{blue}{-\frac{1}{4}}\color{black}{}\cdot -8\geq \color{blue}{-\frac{1}{4}}\color{black}{}\cdot -4x>\color{blue}{-\frac{1}{4}}\color{black}{}\cdot 0&\text{Simplify} \\ 2\color{blue}{\geq}\color{black}{}x\color{blue}{>}\color{black}{}0&\text{Rewrite with }\geq,\: >\text{ signs} \\ 0<x\leq 2&\text{Solution}\end{array}\nonumber$

Let's graph the solution $$0<x\leq 2$$:

Next, we write $$0 < x ≤ 2$$ in interval notation. This is case 2 and the interval is $$(0, 2]$$.

## Linear Inequalities Homework

Graph the inequalities and rewrite the inequalities in interval notation.

##### Exercise $$\PageIndex{1}$$

$$n>-5$$

##### Exercise $$\PageIndex{2}$$

$$-2\geq k$$

##### Exercise $$\PageIndex{3}$$

$$5\geq x$$

##### Exercise $$\PageIndex{4}$$

$$n>4$$

##### Exercise $$\PageIndex{5}$$

$$1\geq k$$

##### Exercise $$\PageIndex{6}$$

$$-5<x$$

Given the graph, write the equivalent inequality and interval notation.

##### Exercise $$\PageIndex{12}$$

Solve the inequality. Graph the solution and write the solution in interval notation.

##### Exercise $$\PageIndex{13}$$

$$\frac{x}{11}\geq 10$$

##### Exercise $$\PageIndex{14}$$

$$2+r<3$$

##### Exercise $$\PageIndex{15}$$

$$8+\frac{n}{3}\geq 6$$

##### Exercise $$\PageIndex{16}$$

$$2>\frac{a-2}{5}$$

##### Exercise $$\PageIndex{17}$$

$$−47 ≥ 8 − 5x$$

##### Exercise $$\PageIndex{18}$$

$$−2(3 + k) < −44$$

##### Exercise $$\PageIndex{19}$$

$$18 < −2(−8 + p)$$

##### Exercise $$\PageIndex{20}$$

$$24 ≥ −6(m − 6)$$

##### Exercise $$\PageIndex{21}$$

$$−r − 5(r − 6) < −18$$

##### Exercise $$\PageIndex{22}$$

$$24 + 4b < 4(1 + 6b)$$

##### Exercise $$\PageIndex{23}$$

$$−5v − 5 < −5(4v + 1)$$

##### Exercise $$\PageIndex{24}$$

$$4 + 2(a + 5) < −2(−a − 4)$$

##### Exercise $$\PageIndex{25}$$

$$−(k − 2) > −k − 20$$

##### Exercise $$\PageIndex{26}$$

$$−2 ≤ \frac{n}{13}$$

##### Exercise $$\PageIndex{27}$$

$$\frac{m}{5}\leq -\frac{6}{5}$$

##### Exercise $$\PageIndex{28}$$

$$11>8+\frac{x}{2}$$

##### Exercise $$\PageIndex{29}$$

$$\frac{v-9}{-4}\leq 2$$

##### Exercise $$\PageIndex{30}$$

$$\frac{6+x}{12}\leq -1$$

##### Exercise $$\PageIndex{31}$$

$$−7n − 10 ≥ 60$$

##### Exercise $$\PageIndex{32}$$

$$5\geq\frac{x}{5}+1$$

##### Exercise $$\PageIndex{33}$$

$$−8(n − 5) ≥ 0$$

##### Exercise $$\PageIndex{34}$$

$$−60 ≥ −4(−6x − 3)$$

##### Exercise $$\PageIndex{35}$$

$$−8(2 − 2n) ≥ −16 + n$$

##### Exercise $$\PageIndex{36}$$

$$−36 + 6x > −8(x + 2) + 4x$$

##### Exercise $$\PageIndex{37}$$

$$3(n + 3) + 7(8 − 8n) < 5n + 5 + 2$$

##### Exercise $$\PageIndex{38}$$

$$−(4 − 5p) + 3 ≥ −2(8 − 5p)$$

##### Exercise $$\PageIndex{39}$$

$$3 ≤ 9 + x ≤ 7$$

##### Exercise $$\PageIndex{40}$$

$$11 < 8 + k ≤ 12$$

##### Exercise $$\PageIndex{41}$$

$$−3 < x − 1 < 1$$

##### Exercise $$\PageIndex{42}$$

$$−4 < 8 − 3m ≤ 11$$

##### Exercise $$\PageIndex{43}$$

$$−16 ≤ 2n − 10 ≤ −22$$

##### Exercise $$\PageIndex{44}$$

$$0\geq\frac{x}{9}\geq -1$$

##### Exercise $$\PageIndex{45}$$

$$-11\leq n-9\leq -5$$

##### Exercise $$\PageIndex{46}$$

$$1\leq\frac{p}{8}\leq 0$$

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