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 ?

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.

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

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

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 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. 5. |2x + 5| = 3x for Examples 1, 2 and 3 The solution of is 5. Reject 1 because it is an extraneous solution. ANSWER

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 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

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

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 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

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

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 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

Solving Absolute Value Inequalities Review

Review of the Steps to Solve a Compound Inequality: ● Example: ● This is a conjunction because the two inequality statements are joined by the word “and”. ● You must solve each part of the inequality. ● The graph of the solution of the conjunction is the intersection of the two inequalities. Both conditions of the inequalities must be met. ● In other words, the solution is wherever the two inequalities overlap. ● If the solution does not overlap, there is no solution.

● Example: ● This is a disjunction because the two inequality statements are joined by the word “or”. ● You must solve each part of the inequality. ● The graph of the solution of the disjunction is the union of the two inequalities. Only one condition of the inequality must be met. ● In other words, the solution will include each of the graphed lines. The graphs can go in opposite directions or towards each other, thus overlapping. ● If the inequalities do overlap, the solution is all reals. Review of the Steps to Solve a Compound Inequality:

“ and’’ Statements can be Written in Two Different Ways ● 1. 8 < m + 6 < 14 ● 2. 8 < m+6 and m+6 < 14 These inequalities can be solved using two methods.

Method One Example : 8 < m + 6 < 14 Rewrite the compound inequality using the word “and”, then solve each inequality. 8 < m + 6 and m + 6 < 14 2 < m m < 8 m >2 and m < 8 2 < m < 8 Graph the solution: 8 2

Example: 8 < m + 6 < 14 To solve the inequality, isolate the variable by subtracting 6 from all 3 parts. 8 < m + 6 < < m < 8 Graph the solution. 8 2 Method Two

‘or’ Statements Example: x - 1 > 2 or x + 3 < -1 x > 3 x < -4 x 3 Graph the solution. 3 -4

Solving an Absolute Value Inequality ● Step 1: Rewrite the inequality as a conjunction or a disjunction. ● If you have a you are working with a conjunction or an ‘and’ statement. Remember: “Less thand” ● If you have a you are working with a disjunction or an ‘or’ statement. Remember: “Greator” ● Step 2: In the second equation you must negate the right hand side and reverse the direction of the inequality sign. ● Solve as a compound inequality.

Example 1: ● |2x + 1| > 7 ● 2x + 1 > 7 or 2x + 1 >7 ● 2x + 1 >7 or 2x + 1 <-7 ● x > 3 or x < -4 This is an ‘or’ statement. (Greator). Rewrite. In the 2 nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. 3 -4

Example 2: ● |x -5|< 3 ● x -5< 3 and x -5< 3 ● x ● x 2 ● 2 < x < 8 This is an ‘and’ statement. (Less thand). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. 8 2