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Solving Inequalities Solving inequalities follows the same procedures as solving equations. There are a few special things to consider with inequalities: We need to look carefully at the inequality sign. We also need to graph the solution set.

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**Review of Inequality Signs**

> greater than < less than greater than or equal less than or equal

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**How to graph the solutions**

> Graph any number greater than. . . open circle, line to the right < Graph any number less than. . . open circle, line to the left Graph any number greater than or equal to. . . closed circle, line to the right Graph any number less than or equal to. . . closed circle, line to the left

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**Solve the inequality: -4 -4 x < 3 x + 4 < 7**

x < 3 Subtract 4 from each side. Keep the same inequality sign. Graph the solution. Open circle, line to the left. 3

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**There is one special case.**

Sometimes you may have to reverse the direction of the inequality sign!! That only happens when you multiply or divide both sides of the inequality by a negative number.

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**Example: -3y > 18 Solve: -3y + 5 >23 -5 -5**

-3y > 18 y < -6 Subtract 5 from each side. Divide each side by negative 3. Reverse the inequality sign. Graph the solution. Open circle, line to the left. -6

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EXAMPLE 1 Graph simple inequalities a. Graph x < 2. The solutions are all real numbers less than 2. An open dot is used in the graph to indicate 2 is not a solution.

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EXAMPLE 1 Graph simple inequalities b. Graph x ≥ –1. The solutions are all real numbers greater than or equal to –1. A solid dot is used in the graph to indicate –1 is a solution.

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EXAMPLE 2 Graph compound inequalities b. Graph x ≤ –2 or x > 1. The solutions are all real numbers that are less than or equal to –2 or greater than 1.

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EXAMPLE 2 Graph compound inequalities a. Graph –1 < x < 2. The solutions are all real numbers that are greater than –1 and less than 2.

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GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 1. x > –5 The solutions are all real numbers greater than 5. An open dot is used in the graph to indicate –5 is not a solution.

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GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 2. x ≤ 3 The solutions are all real numbers less than or equal to 3. A closed dot is used in the graph to indicate 3 is a solution.

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GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 3. –3 ≤ x < 1 The solutions are all real numbers that are greater than or equalt to –3 and less than 1.

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GUIDED PRACTICE for Examples 1 and 2 Graph the inequality. 4. x < 1 or x ≥ 2 The solutions are all real numbers that are less than 1 or greater than or equal to 2.

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EXAMPLE 3 Solve an inequality with a variable on one side Fair You have $50 to spend at a county fair. You spend $20 for admission. You want to play a game that costs $1.50. Describe the possible numbers of times you can play the game. SOLUTION STEP 1 Write a verbal model. Then write an inequality.

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**Solve an inequality with a variable on one side**

EXAMPLE 3 Solve an inequality with a variable on one side An inequality is g ≤ 50. STEP 2 Solve the inequality. g ≤ 50 Write inequality. 1.5g ≤ 30 Subtract 20 from each side. g ≤ 20 Divide each side by 1.5. ANSWER You can play the game 20 times or fewer.

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**Solve an inequality with a variable on both sides**

EXAMPLE 4 Solve an inequality with a variable on both sides Solve 5x + 2 > 7x – 4. Then graph the solution. 5x + 2 > 7x – 4 Write original inequality. – 2x + 2 > – 4 Subtract 7x from each side. – 2x > – 6 Subtract 2 from each side. Divide each side by –2 and reverse the inequality. x < 3 ANSWER The solutions are all real numbers less than 3. The graph is shown below.

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GUIDED PRACTICE for Examples 3 and 4 Solve the inequality. Then graph the solution. 5. 4x + 9 < 25 7. 5x – 7 ≤ 6x x < 4 ANSWER x > – 7 ANSWER 6. 1 – 3x ≥ –14 8. 3 – x > x – 9 x ≤ 5 ANSWER x < 6 ANSWER

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**Solving Inequalities -4x + 2 > 10 -4x > 8 x < -2**

To graph the solution set, circle the boundary and shade according to the inequality. Use an open circle for < or > and closed circles for ≤ or ≥. -2 -1

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**Solving Inequalities 3b - 2(b - 5) < 2(b + 4)**

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**Solve an “and” compound inequality**

EXAMPLE 5 Solve an “and” compound inequality Solve – 4 < 6x – 10 ≤ 14. Then graph the solution. – 4 < 6x – 10 ≤ 14 Write original inequality. – < 6x – ≤ Add 10 to each expression. 6 < 6x ≤ 24 Simplify. 1 < x ≤ 4 Divide each expression by 6. ANSWER The solutions are all real numbers greater than 1 and less than or equal to 4. The graph is shown below.

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GUIDED PRACTICE for Examples 5,6, and 7 Solve the inequality. Then graph the solution. 9. –1 < 2x + 7 < 19 ANSWER The solutions are all real numbers greater than – 4 and less than 6. –4 < x < 6

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**Solve an “or” compound inequality**

EXAMPLE 6 Solve an “or” compound inequality Solve 3x + 5 ≤ 11 or 5x – 7 ≥ 23 . Then graph the solution. SOLUTION A solution of this compound inequality is a solution of either of its parts. First Inequality Second Inequality 3x + 5 ≤ 11 Write first inequality. 5x – 7 ≥ 23 Write second inequality. 3x ≤ 6 5x ≥ 30 Subtract 5 from each side. Add 7 to each side. x ≥ 6 x ≤ 2 Divide each side by 3. Divide each side by 5.

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EXAMPLE 6 Solve an “or” compound inequality ANSWER The graph is shown below. The solutions are all real numbers less than or equal to 2 or greater than or equal to 6.

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EXAMPLE 7 Write and use a compound inequality Biology A monitor lizard has a temperature that ranges from 18°C to 34°C. Write the range of temperatures as a compound inequality. Then write an inequality giving the temperature range in degrees Fahrenheit.

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**Write and use a compound inequality**

EXAMPLE 7 Write and use a compound inequality SOLUTION The range of temperatures C can be represented by the inequality 18 ≤ C ≤ 34. Let F represent the temperature in degrees Fahrenheit. 18 ≤ C ≤ 34 Write inequality. 18 ≤ ≤ 34 5 9 (F – 32) Substitute for C. 9 5 (F – 32) Multiply each expression by , the reciprocal of 9 5 32.4 ≤ F – 32 ≤ 61.2 64.4 ≤ F ≤ 93.2 Add 32 to each expression.

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EXAMPLE 7 Write and use a compound inequality ANSWER The temperature of the monitor lizard ranges from 64.4°F to 93.2°F.

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GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 10. –8 ≤ –x – 5 ≤ 6 The solutions are all real numbers greater than and equal to – 11 and less than and equal to 3. ANSWER –11 ≤ x ≤ 3

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GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 11. x + 4 ≤ 9 or x – 3 ≥ 7 ANSWER The graph is shown below. The solutions are all real numbers. less than or equal to 5 or greater than or equal to 10. x ≤ 5 or x ≥ 10

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GUIDED PRACTICE for Examples 5,6 and 7 Solve the inequality. Then graph the solution. 12. 3x – 1< –1 or 2x + 5 ≥ 11 x < 0 or x ≥ 3 less than 0 or greater than or equal to 3. ANSWER The graph is shown below. The solutions are all real numbers.

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GUIDED PRACTICE for Examples 5,6 and 7 13. WHAT IF? In Example 7, write a compound inequality for a lizard whose temperature ranges from 15°C to 30°C. Then write an inequality giving the temperature range in degrees Fahrenheit. ANSWER 15 ≤ C ≤ 30 or 59 ≤ F ≤ 86

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