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**Solve a simple absolute value equation**

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

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EXAMPLE 1 Solve a simple absolute value equation ANSWER 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.

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**Solve an absolute value equation**

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

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**Solve an absolute value equation**

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

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**Check for extraneous solutions**

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

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**Check for extraneous solutions**

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

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**Solve the equation. Check for extraneous solutions.**

GUIDED PRACTICE for Examples 1, 2 and 3 Solve the equation. Check for extraneous solutions. 1. | x | = 5 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 – 1 1 2 3 4 5 6 7 – 5 – 6 – 7

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**Solve the equation. Check for extraneous solutions.**

GUIDED PRACTICE for Examples 1, 2 and 3 Solve the equation. Check for extraneous solutions. 2. |x – 3| = 10 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 – 1 1 2 3 4 5 6 7 – 5 – 6 – 7 8 9 10 11 12 13

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

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

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

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GUIDED PRACTICE for Examples 1, 2 and 3 Solve the equation. Check for extraneous solutions. 6. |4x – 1| = 2x + 9 ANSWER The solutions are – and 5. 3 1

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