Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

In this activity, we’re going to solve inequalities. Work with a partner to solve each inequality given. Then match it with the number line representing its solution.

As you complete the activity, discuss with your partner how you know each number line is the solution to its inequality.

When you think you’ve found the correct number line for an inequality, substitute a test point that is on the highlighted part of the number line into the inequality. Is it true?

Try a test point that is not highlighted. Is the inequality false?

1. $x - 4 \leq - 6$	Graph A
2. $x + 6 \geq 8$	Graph B
3. $4 x \leq 8$	Graph C
4. $- \frac{1}{2} x \leq 1$	Graph D

Use the table to choose the matching graph to the inequality.

1.

$x - 4 \leq - 6$

Graph C.

2.

$x + 6 \geq 8$

Graph A.

3.

$4 x \leq 8$

Graph D.

4.

$- \frac{1}{2} x \leq 1$

Graph B.

For each of the inequalities above, match it to the property you used to solve.

5. $x - 4 \leq - 6$	A. Multiplication Property of Inequality
6. $x + 6 \geq 8$	B. Addition Property of Inequality
7. $4 x \leq 8$	C. Division Property of Inequality
8. $- \frac{1}{2} x \leq 1$	D. Subtraction Property of Inequality

5.

$x - 4 \leq - 6$

Addition Property of Inequality.

6.

$x + 6 \geq 8$

Subtraction Property of Inequality.

7.

$4 x \leq 8$

Select two properties that could be used to solve the inequality

Multiplication Property of Inequality and Division Property of Inequality

8.

$- \frac{1}{2} x \leq 1$

Select two properties that could be used to solve the inequality

Multiplication Property of Inequality and Division Property of Inequality

9.

Is there anything unique about the method used to solve $- \frac{1}{2} x \leq 1$ ? Explain.

Compare your answer:

Yes. The inequality contains a less than or equal to symbol $(\leq)$ . However, in the solution there is a greater than or equal to symbol $(\geq)$ . When using the Multiplication and Division Properties of Inequality, you must reverse the symbol when you multiply or divide by a negative number.

Self Check

Solve the inequality. Which number line represents the solution?

$–\;\frac13m\;\leq\;5$

Additional Resources

Solving Linear Inequalities

A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. So, when we solve linear equations, we are able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:

First example shows negative 4 is less than 2. The quantity of negative 4 minus 5 is less than the quantity of 2 minus 5. This yields negative 9 less than negative 3, which is true. The second example shows negative 4 is less than 2. The quantity of negative 4 plus 7 is less than the quantity of 2 plus 7. This results in 3 less than 9, which is true.

Notice that the inequality sign stayed the same.

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

Addition and Subtraction Properties of Inequality

For any numbers $a$ , $b$ , and $c$ , if $a < b$ , then

$a + c < b + c$

$a - c < b - c$

For any numbers $a$ , $b$ , and $c$ , if $a > b$ , then

$a + c > b + c$

$a - c > b - c$

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

What happens to an inequality when we divide or multiply both sides by a constant?

Let’s first multiply and divide both sides by a positive number.

First example shows 10 is less than 15. The quantity of 10 times 5 is less than the quantity of 15 times 5. This yields 50 less than 75, which is true. The second example shows 10 is less than 15. The quantity of 10 divided by 5 is less than the quantity of 15 divided by 5. This results in 2 less than 3, which is true.

The inequality signs stayed the same.

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

First example shows 10 is less than 15. The quantity of 10 times negative 5 is blank the quantity of 15 times negative 5. This yields negative 50 blank negative 75. Negative 50 is greater than negative 75. The second example shows 10 is less than 15. The quantity of 10 divided by negative 5 is blank the quantity of 15 divided by negative 5. This results in negative 2 blank negative 3. Negative 2 is greater than negative 3.

Notice that when we filled in the inequality signs, 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.

This gives us the Multiplication and Division Properties of Inequality.

Multiplication and Division Properties of Inequality

For any numbers $a$ , $b$ , and $c$ ,

multiply or divide by a positive

if $a < b$ and $c > 0$ , then $a c < b c$ and $\frac{a}{c} < \frac{b}{c}$

if $a > b$ and $c > 0$ , then $a c > b c$ and $\frac{a}{c} > \frac{b}{c}$

Multiply or divide by a negative

if $a < b$ and $c < 0$ , then $a c > b c$ and $\frac{a}{c} > \frac{b}{c}$

if $a > b$ and $c < 0$ , then $a c < b c$ and $\frac{a}{c} < \frac{b}{c}$

When we divide or multiply an inequality by a:

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

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

$x > a$ has the same meaning as $a < x$ .

Think about it as “If Xander is taller than Andy, then Andy is shorter than Xander.”

In Examples 1 and 2: solve the inequality, graph the solution on the number line, and write the solution in interval notation.

Example 1

$x - \frac{3}{8} \leq \frac{3}{4}$

Solution

Step 1 - Add $\frac{3}{8}$ to both sides of the inequality.

$x - \frac{3}{8} + \frac{3}{8} \leq \frac{3}{4} + \frac{3}{8}$

Step 2 - Simplify.

$x \leq \frac{9}{8}$

Step 3 - Graph the solution on the number line.

$A number line with a closed bracket is graphed at the fractional value of nine-eighths. A bold arrowed line extending left, shades the numbers less than nine-eighths.$

Step 4 - Write the solution in interval notation.

$(- \infty, \frac{9}{8}]$

Example 2

$- 15 < \frac{3}{5} z$

Solution

Step 1 - Multiply both sides of the inequality by $\frac{5}{3}$ . Since $\frac{5}{3}$ is positive, the inequality stays the same.

$(\frac{5}{3}) (- 15) < (\frac{5}{3}) (\frac{3}{5} z)$

Step 2 - Simplify.

$- 25 < z$

Step 3 - Rewrite with the variable on the left.

$z > - 25$

Step 4 - Graph the solution on the number line.

Step 5 - Write the solution in interval notation.

$(- 25, \infty)$

Try it

Try It: Solving Linear Inequalities

Solve the following linear inequalities:

1.

$9 y < 54$

Compare your answer:

Step 1 - Divide both sides of the inequality by 9; since 9 is positive, the inequality stays the same.

$\frac{9 y}{9} < \frac{54}{9}$

Step 2 - Simplify.

$y < 6$

Step 3 - Graph the solution on the number line.

Step 4 - Write the solution in interval notation.

$(- \infty, 6)$

2.

$- 13 m \geq 65$

Compare your answer:

Step 1 - Divide both sides of the inequality by $- 13$ . Since $- 13$ is negative, the inequality reverses.

$\frac{- 13 m}{- 13} \leq \frac{65}{- 13}$

Step 2 - Simplify.

$m \leq - 5$

Step 3 - Graph the solution on the number line.

Step 4 - Write the solution in interval notation.

$(- \infty, - 5)$

2.10.5 Solving Inequalities

Activity

Additional Resources

Solving Linear Inequalities

Try It: Solving Linear Inequalities