Chapter 1 Section 7

EXAMPLE 1 Solve a simple absolute value equation Solve |x – 5| = 7. Graph the solution. SOLUTION | x – 5 | = 7 x – 5 = – 7 or x – 5 = 7 x = 5 – 7 or x = x = –2 or x = 12 Write original equation. Write equivalent equations. Solve for x. Simplify.

EXAMPLE 1 The solutions are –2 and 12. These are the values of x that are 7 units away from 5 on a number line. The graph is shown below. ANSWER Solve a simple absolute value equation

EXAMPLE 2 Solve an absolute value equation | 5x – 10 | = 45 5x – 10 = 45 or 5x – 10 = –45 5x = 55 or 5x = –35 x = 11 or x = –7 Write original equation. Expression can equal 45 or –45. Add 10 to each side. Divide each side by 5. Solve |5x – 10 | = 45. SOLUTION

EXAMPLE 2 Solve an absolute value equation The solutions are 11 and –7. Check these in the original equation. ANSWER Check: | 5x – 10 | = 45 | 5(11) – 10 | = 45 ? |45| = 45 ? 45 = 45 | 5x – 10 | = 45 | 5(–7) – 10 | = 45 ? 45 = 45 | – 45| = 45 ?

EXAMPLE 3 | 2x + 12 | = 4x 2x + 12 = 4x or 2x + 12 = – 4x 12 = 2x or 12 = –6x 6 = x or –2 = x Write original equation. Expression can equal 4x or – 4 x Add –2x to each side. Solve |2x + 12 | = 4x. Check for extraneous solutions. SOLUTION Solve for x. Check for extraneous solutions

EXAMPLE 3 | 2x + 12 | = 4x | 2(–2) +12 | = 4(–2) ? |8| = – 8 ? 8 = –8 Check the apparent solutions to see if either is extraneous. Check for extraneous solutions | 2x + 12 | = 4x | 2(6) +12 | = 4(6) ? |24| = 24 ? 24 = 24 The solution is 6. Reject –2 because it is an extraneous solution. ANSWER CHECK

GUIDED PRACTICE Solve the equation. Check for extraneous solutions. 1. | x | = 5 for Examples 1, 2 and 3 The solutions are –5 and 5. These are the values of x that are 5 units away from 0 on a number line. The graph is shown below. ANSWER – 3 – 4 – 2 – – 5 – 6 – 7 5 5

GUIDED PRACTICE Solve the equation. Check for extraneous solutions. 2. |x – 3| = 10 for Examples 1, 2 and 3 The solutions are –7 and 13. These are the values of x that are 10 units away from 3 on a number line. The graph is shown below. ANSWER – 3 – 4 – 2 – – 5 – 6 –

GUIDED PRACTICE Solve the equation. Check for extraneous solutions. 3. |x + 2| = 7 for Examples 1, 2 and 3 The solutions are –9 and 5. These are the values of x that are 7 units away from – 2 on a number line. ANSWER

GUIDED PRACTICE Solve the equation. Check for extraneous solutions. 4. |3x – 2| = 13 for Examples 1, 2 and 3 ANSWER The solutions are 5 and.

GUIDED PRACTICE Solve the equation. Check for extraneous solutions. 5. |2x + 5| = 3x for Examples 1, 2 and 3 The solution of is 5. Reject 1 because it is an extraneous solution. ANSWER

GUIDED PRACTICE Solve the equation. Check for extraneous solutions. 6. |4x – 1| = 2x + 9 for Examples 1, 2 and 3 ANSWER The solutions are – and

EXAMPLE 4 Solve an inequality of the form |ax + b| > c Solve |4x + 5| > 13. Then graph the solution. SOLUTION First Inequality Second Inequality 4x + 5 < –134x + 5 > 13 4x < –184x > 8 x < – 9 2 x > 2 Write inequalities. Subtract 5 from each side. Divide each side by 4. The absolute value inequality is equivalent to 4x

EXAMPLE 4 ANSWER Solve an inequality of the form |ax + b| > c The solutions are all real numbers less than or greater than 2. The graph is shown below. – 9 2

GUIDED PRACTICE for Example 4 Solve the inequality. Then graph the solution. 7. |x + 4| ≥ 6 x 2 The graph is shown below. ANSWER

GUIDED PRACTICE for Example 4 Solve the inequality. Then graph the solution. 8. |2x –7|>1 ANSWER x 4 The graph is shown below.

GUIDED PRACTICE for Example 4 Solve the inequality. Then graph the solution. 9. |3x + 5| ≥ 10 ANSWER x The graph is shown below.

EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c A professional baseball should weigh ounces, with a tolerance of ounce. Write and solve an absolute value inequality that describes the acceptable weights for a baseball. Baseball SOLUTION Write a verbal model. Then write an inequality. STEP 1

EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c STEP 2Solve the inequality. Write inequality. Write equivalent compound inequality. Add to each expression. |w – 5.125| ≤ – ≤ w – ≤ ≤ w ≤ 5.25 So, a baseball should weigh between 5 ounces and 5.25 ounces, inclusive. The graph is shown below. ANSWER

EXAMPLE 6 The thickness of the mats used in the rings, parallel bars, and vault events must be between 7.5 inches and 8.25 inches, inclusive. Write an absolute value inequality describing the acceptable mat thicknesses. Gymnastics SOLUTION STEP 1 Calculate the mean of the extreme mat thicknesses. Write a range as an absolute value inequality

EXAMPLE 6 Mean of extremes = = Find the tolerance by subtracting the mean from the upper extreme. STEP 2 Tolerance = 8.25 – Write a range as an absolute value inequality = 0.375

EXAMPLE 6 STEP 3 Write a verbal model. Then write an inequality. A mat is acceptable if its thickness t satisfies |t – 7.875| ≤ ANSWER Write a range as an absolute value inequality

GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. 10. |x + 2| < 6 The solutions are all real numbers less than – 8 or greater than 4. The graph is shown below. ANSWER –8 < x < 4

GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. 11. |2x + 1| ≤ 9 The solutions are all real numbers less than –5 or greater than 4. The graph is shown below. ANSWER –5 ≤ x ≤ 4

GUIDED PRACTICE for Examples 5 and |7 – x| ≤ 4 Solve the inequality. Then graph the solution. 3 ≤ x ≤ 11 ANSWER The solutions are all real numbers less than 3 or greater than 11. The graph is shown below.

GUIDED PRACTICE for Examples 5 and Gymnastics: For Example 6, write an absolute value inequality describing the unacceptable mat thicknesses. A mat is unacceptable if its thickness t satisfies |t – 7.875| > ANSWER