2.7 Solve Linear Inequalities

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 2.7. Please note that the following convention is used: thick arrows point in the positive direction and thin arrows point in the negative direction.

The graph of the inequality $x\ge 3$ 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 2.8.

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.

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 $\left(3,\infty \right).$ The symbol $\infty$ is read as ‘infinity’. It is not an actual number. Figure 2.9 shows both the number line and the interval notation.

The inequality $x\le 1$ means all numbers less than or equal to 1. There is no lower end to those numbers. We write $x\le 1$ in interval notation as $(\text{−}\infty ,1]$. The symbol $\text{−}\infty$ is read as ‘negative infinity’. Figure 2.10 shows both the number line and interval notation.

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 2.11.

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.

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.

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

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

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:

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

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).

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.

$$\mathbf{\text{The inequality signs stayed the same.}}$$

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.

$$\mathbf{\text{The inequality signs reversed their direction.}}$$

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.

When we divide or multiply an inequality by a:

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

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.

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.

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 2.7 shows some common phrases that indicate inequalities.

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 $\frac{a}{c}<\frac{b}{c}$ and $ac>bc$.
  if $a>b$ and $c>0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$.
  if $a<b$ and $c<0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$.
  if $a>b$ and $c<0$, then $\frac{a}{c}<\frac{b}{c}$ and $ac<bc$.
  
• When we divide or multiply an inequality by a:
  • positive number, the inequality stays the same.
  • negative number, the inequality reverses.

Section Exercises

Practice Makes Perfect

Graph Inequalities on the Number Line

In the following exercises, graph each inequality on the number line.

ⓐ $x\le -2$
ⓑ $x>-1$ⓒ $x<0$

ⓐ $x>1$ⓑ $x<-2$
ⓒ $x\ge \mathrm{-3}$

1. ⓐ
   

   

2. ⓑ
   

   

3. ⓒ
   

   

ⓐ $x\ge \mathrm{-3}$ⓑ $x<4$ⓒ $x\le -2$

ⓐ $x\le 0$ⓑ $x>-4$ⓒ $x\ge \mathrm{-1}$

1. ⓐ
   

   

2. ⓑ
   

   

3. ⓒ
   

   

In the following exercises, graph each inequality on the number line and write in interval notation.

ⓐ $x<-2$ⓑ $x\ge \mathrm{-3.5}$ⓒ $x\le \frac{2}{3}$

ⓐ $x>3$ⓑ $x\le -0.5$ⓒ $x\ge \frac{1}{3}$

1. ⓐ
   

   

2. ⓑ
   

   

3. ⓒ
   

   

ⓐ $x\ge \mathrm{-4}$ⓑ $x<2.5$ⓒ $x>-\frac{3}{2}$

ⓐ $x\le 5$ⓑ $x\ge \mathrm{-1.5}$ⓒ $x<-\frac{7}{3}$

1. ⓐ
   

   

2. ⓑ
   

   

3. ⓒ
   

   

Solve Inequalities using the Subtraction and Addition Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$$n-11<33$$

$$m-45\le 62$$

$$u+25>21$$

$$v+12>3$$

$$a+\frac{3}{4}\ge \frac{7}{10}$$

$$b+\frac{7}{8}\ge \frac{1}{6}$$

$$f-\frac{13}{20}<-\frac{5}{12}$$

$$g-\frac{11}{12}<-\frac{5}{18}$$

Solve Inequalities using the Division and Multiplication Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$$8x>72$$

$$6y<48$$

$$7r\le 56$$

$$9s\ge 81$$

$$\mathrm{-5}u\ge 65$$

$$\mathrm{-8}v\le 96$$

$$\mathrm{-9}c<126$$

$$\mathrm{-7}d>105$$

$$20>\frac{2}{5}h$$

$$40<\frac{5}{8}k$$

$$\frac{7}{6}j\ge 42$$

$$\frac{9}{4}g\le 36$$

$$\frac{a}{\mathrm{-3}}\le 9$$

$$\frac{b}{\mathrm{-10}}\ge 30$$

$$\mathrm{-25}<\frac{p}{\mathrm{-5}}$$

$$\mathrm{-18}>\frac{q}{\mathrm{-6}}$$

$$9t\ge \mathrm{-27}$$

$$7s<-28$$

$$\frac{2}{3}y>-36$$

$$\frac{3}{5}x\le -45$$

Solve Inequalities That Require Simplification

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$$4v\ge 9v-40$$

$$5u\le 8u-21$$

$$13q<7q-29$$

$$9p>14p-18$$

$$12x+3(x+7)>10x-24$$

$$9y+5(y+3)<4y-35$$

$$6h-4(h-1)\le 7h-11$$

$$4k-(k-2)\ge 7k-26$$

$$8m-2\left(14-m\right)\ge \text{}7\left(m-4\right)+3m$$

$$6n-12\left(3-n\right)\le 9\left(n-4\right)+9n$$

$$\frac{3}{4}b-\frac{1}{3}b<\frac{5}{12}b-\frac{1}{2}$$

$$9u+5(2u-5)\ge 12(u-1)+7u$$

$$\frac{2}{3}g-\frac{1}{2}(g-14)\le \frac{1}{6}(g+42)$$

$$\frac{5}{6}a-\frac{1}{4}a>\frac{7}{12}a+\frac{2}{3}$$

$$\frac{4}{5}h-\frac{2}{3}\left(h-9\right)\ge \frac{1}{15}\left(2h+90\right)$$

$$12v+3(4v-1)\le 19(v-2)+5v$$

Mixed practice

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$$15k\le -40$$

$$35k\ge \mathrm{-77}$$

$$23p-2(6-5p)>3(11p-4)$$

$$18q-4(10-3q)<5(6q-8)$$

$$-\frac{9}{4}x\ge -\frac{5}{12}$$

$$-\frac{21}{8}y\le -\frac{15}{28}$$

$$c+34<-99$$

$$d+29>-61$$

$$\frac{m}{18}\ge \mathrm{-4}$$

$$\frac{n}{13}\le -6$$

Translate to an Inequality and Solve

In the following exercises, translate and solve.Then write the solution in interval notation and graph on the number line.

Fourteen times d is greater than 56.

Ninety times c is less than 450.

Eight times z is smaller than $\mathrm{-40}$.

Ten times y is at most $\mathrm{-110}$.

Three more than h is no less than 25.

Six more than k exceeds 25.

Ten less than w is at least 39.

Twelve less than x is no less than 21.

Negative five times r is no more than 95.

Negative two times s is lower than 56.

Nineteen less than b is at most $\mathrm{-22}$.

Fifteen less than a is at least $\mathrm{-7}$.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Chapter 2 Review Exercises

Solve Linear Inequalities

Graph Inequalities on the Number Line

In the following exercises, graph each inequality on the number line.

ⓐ $x\le 4$
ⓑ $x>-2$
ⓒ $x<1$

ⓐ $x>0$
ⓑ $x<-3$
ⓒ $x\ge \mathrm{-1}$

1. ⓐ
   

   

2. ⓑ
   

   

3. ⓒ
   

   

In the following exercises, graph each inequality on the number line and write in interval notation.

ⓐ $x<-1$
ⓑ $x\ge \mathrm{-2.5}$
ⓒ $x\le \frac{5}{4}$

ⓐ $x>2$
ⓑ $x\le -1.5$
ⓒ $x\ge \frac{5}{3}$

1. ⓐ
   

   

2. ⓑ
   

   

3. ⓒ
   

   

Solve Inequalities using the Subtraction and Addition Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$$n-12\le 23$$

$$m+14\le 56$$

$$a+\frac{2}{3}\ge \frac{7}{12}$$

$$b-\frac{7}{8}\ge -\frac{1}{2}$$

Solve Inequalities using the Division and Multiplication Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$$9x>54$$

$$\mathrm{-12}d\le 108$$

$$\frac{5}{6}j<-60$$

$$\frac{q}{-2}\ge -24$$

Solve Inequalities That Require Simplification

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$$6p>15p-30$$

$$9h-7(h-1)\le 4h-23$$

$$5n-15(4-n)<10(n-6)+10n$$

$$\frac{3}{8}a-\frac{1}{12}a>\frac{5}{12}a+\frac{3}{4}$$

Translate to an Inequality and Solve

In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.

Five more than z is at most 19.

Three less than c is at least 360.

Nine times n exceeds 42.

Negative two times a is no more than 8.

Chapter 2 Practice Test

In the following exercises, solve each equation.

In the following exercises, graph on the number line and write in interval notation.

In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

In the following exercises, translate to an equation or inequality and solve.