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EXAMPLE 1

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 .

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EXAMPLE 1

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

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

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

  4. ? | 5(–7)– 10 |= 45 ? | 5(11)– 10 |= 45 ? | –45|= 45 ? 45= 45 45= 45 |45|= 45 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. EXAMPLE 3 Check for extraneous solutions Solve|2x + 12 | = 4x.Check for extraneous solutions. SOLUTION | 2x + 12 | = 4x Write original equation. 2x + 12 = 4xor 2x + 12 = – 4x Expression can equal 4xor – 4 x 12= 2xor 12= –6x Add –2x to each side. 6 = x or –2= x Solve for x.

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

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

  8. ANSWER The solutions are –7 and 13. These are the values of xthat are 10units away from 3on a number line. The graph is shown below. 10 10 – 2 0 1 2 3 4 5 6 10 11 – 6 12 – 3 – 1 7 13 – 7 8 9 – 5 – 4 for Examples 1, 2 and 3 GUIDED PRACTICE Solve the equation. Check for extraneous solutions. 2. |x – 3| = 10

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

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

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

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

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