2ObjectiveSolve equations in one variable that contain absolute-value expressions.
3The absolute-value of a number is that numbers distance from zero on a number line. For example, |–5| = 5.5 units654321123456Both 5 and –5 are a distance of 5 units from 0, so both 5 and –5 have an absolute value of 5.To write this using algebra, you would write |x| = 5. This equation asks, “What values of x have an absolute value of 5?” The solutions are 5 and –5. Notice this equation has two solutions.
5To solve absolute-value equations, perform inverse operations to isolate the absolute-value expression on one side of the equation. Then you must consider two cases.
6Example 1A: Solving Absolute-Value Equations Solve each equation. Check your answer.|x| = 12Think: What numbers are 12 units from 0?|x| = 12Case 1x = 12Case 2x = –12Rewrite the equation as two cases.The solutions are 12 and –12.Check|x| = 12|12||12| 12
7Example 1B: Solving Absolute-Value Equations Since |x + 7| is multiplied by 3, divide both sides by 3 to undo the multiplication.Think: What numbers are 8 units from 0?|x + 7| = 8Case 1x + 7 = 8Case 2x + 7 = –8– 7 –7– 7 – 7x = 1x = –15Rewrite the equations as two cases. Since 7 is added to x subtract 7 from both sides of each equation.The solutions are 1 and –15.
9Check It Out! Example 1aSolve each equation. Check your answer.|x| – 3 = 4Since 3 is subtracted from |x|, add 3 to both sides.|x| – 3 = 4|x| = 7Think: what numbers are 7 units from 0?Case 1x = 7Case 2–x = 7–1(–x) = –1(7)x = –7x = 7Rewrite the case 2 equation by multiplying by –1 to change the minus x to a positive..The solutions are 7 and –7.
10Check It Out! Example 1a Continued Solve the equation. Check your answer.|x| 3 = 4The solutions are 7 and 7.Check|x| 3 = 47 |7| | 7| 7
11Check It Out! Example 1bSolve the equation. Check your answer.|x 2| = 8Think: what numbers are 8 units from 0?|x 2| = 8Case 1x 2 = 8x = 10x = 6Case 2x 2 = 8Rewrite the equations as two cases. Since 2 is subtracted from x add 2 to both sides of each equation.The solutions are 10 and 6.
12Check It Out! Example 1b Continued Solve the equation. Check your answer.|x 2| = 8The solutions are 10 and 6.Check|x 2| = 810 2| 8|10 2| 8| 6 + (2)| 8
13Not all absolute-value equations have two solutions Not all absolute-value equations have two solutions. If the absolute-value expression equals 0, there is one solution. If an equation states that an absolute-value is negative, there are no solutions.
14Example 2A: Special Cases of Absolute-Value Equations Solve the equation. Check your answer.8 = |x + 2| 8Since 8 is subtracted from |x + 2|, add 8 to both sides to undo the subtraction.8 = |x + 2| 80 = |x +2|There is only one case. Since 2 is added to x, subtract 2 from both sides to undo the addition.0 = x + 2 22 = x
15Example 2A ContinuedSolve the equation. Check your answer.8 = |x +2| 8Solution is x = 2Check8 =|x + 2| 8To check your solution, substitute 2 for x in your original equation.8 88 |2 + 2| 88 |0| 8 8
16Example 2B: Special Cases of Absolute-Value Equations Solve the equation. Check your answer.3 + |x + 4| = 0Since 3 is added to |x + 4|, subtract 3 from both sides to undo the addition.3 + |x + 4| = 0 3|x + 4| = 3Absolute values cannot be negative.This equation has no solution.
17Remember!Absolute value must be nonnegative because it represents distance.
18Check It Out! Example 2aSolve the equation. Check your answer.2 |2x 5| = 7Since 2 is added to |2x 5|, subtract 2 from both sides to undo the addition.2 |2x 5| = 7 2 |2x 5| = 5Since |2x 5| is multiplied by a negative 1, divide both sides by negative 1.|2x 5| = 5Absolute values cannot be negative.This equation has no solution.
19Check It Out! Example 2bSolve the equation. Check your answer.6 + |x 4| = 6Since 6 is subtracted from |x 4|, add 6 to both sides to undo the subtraction.6 + |x 4| = 6|x 4| = 0x 4 = 0x = 4There is only one case. Since 4 is subtracted from x, add 4 to both sides to undo the addition.
20Check It Out! Example 2b Continued Solve the equation. Check your answer.6 + |x 4| = 6The solution is x = 4.6 + |x 4| = 6To check your solution, substitute 4 for x in your original equation.6 + |4 4| 66 +|0| 6 66 6