H Algebra 2/Trig · Unit 1
Isolate the absolute value, split into two equations when the right side is not negative, and catch no-solution cases.
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Split into two equations
|A| = k means A = k or A = −k, but only when k ≥ 0.
Isolate first
If extra terms sit outside the absolute value, move them before you split.
No solution
If the isolated absolute value equals a negative number, there is no solution.
Check
Plug both candidates back in. A true absolute-value equation should match on both sides.
Already isolated. Two solutions.
The formula is
|A| = k means A = k or A = −k (when k ≥ 0).
|x − 4| = 6
x − 4 = 6 or x − 4 = −6
First equation.
x − 4 = 6
x = 10
Second equation.
x − 4 = −6
x = −2
Answer: x = 10 or x = −2
The formula is
|A| = k means A = k or A = −k (when k ≥ 0).
|x + 2| = 5
x + 2 = 5 or x + 2 = −5
First equation.
x + 2 = 5
x = 3
Second equation.
x + 2 = −5
x = −7
Answer: x = 3 or x = −7
The formula is
|A| = k means A = k or A = −k (when k ≥ 0).
|x − 3| = 7
x − 3 = 7 or x − 3 = −7
First equation.
x − 3 = 7
x = 10
Second equation.
x − 3 = −7
x = −4
Answer: x = 10 or x = −4
The formula is
|A| = k means A = k or A = −k (when k ≥ 0).
|2x + 1| = 9
2x + 1 = 9 or 2x + 1 = −9
First equation.
2x + 1 = 9
x = 4
Second equation.
2x + 1 = −9
x = −5
Answer: x = 4 or x = −5
The formula is
|A| = k means A = k or A = −k (when k ≥ 0).
|3x − 4| = 8
3x − 4 = 8 or 3x − 4 = −8
First equation.
3x − 4 = 8
x = 4
Second equation.
3x − 4 = −8
x = −43
Answer: x = 4 or x = −43
The formula is
|A| = k means A = k or A = −k (when k ≥ 0).
|2x − 5| = 1
2x − 5 = 1 or 2x − 5 = −1
First equation.
2x − 5 = 1
x = 3
Second equation.
2x − 5 = −1
x = 2
Answer: x = 3 or x = 2
Isolate the absolute value, then split.
1|x + 2| + 5 = 11
Isolate the absolute value.
1|x + 2| = 6
The formula is
|A| = k means A = k or A = −k.
x + 2 = 6 or x + 2 = −6
x = 4 or x = −8
Answer: x = 4 or x = −8
1|x − 4| − 3 = 5
Isolate the absolute value.
1|x − 4| = 8
The formula is
|A| = k means A = k or A = −k.
x − 4 = 8 or x − 4 = −8
x = 12 or x = −4
Answer: x = 12 or x = −4
3|x − 1| = 12
Isolate the absolute value.
|x − 1| = 4
The formula is
|A| = k means A = k or A = −k.
x − 1 = 4 or x − 1 = −4
x = 5 or x = −3
Answer: x = 5 or x = −3
2|x + 3| − 4 = 10
Isolate the absolute value.
2|x + 3| = 14
|x + 3| = 7
The formula is
|A| = k means A = k or A = −k.
x + 3 = 7 or x + 3 = −7
x = 4 or x = −10
Answer: x = 4 or x = −10
1|4x − 1| − 6 = 9
Isolate the absolute value.
1|4x − 1| = 15
The formula is
|A| = k means A = k or A = −k.
4x − 1 = 15 or 4x − 1 = −15
x = 4 or x = −72
Answer: x = 4 or x = −72
5|2x + 3| = 20
Isolate the absolute value.
|2x + 3| = 4
The formula is
|A| = k means A = k or A = −k.
2x + 3 = 4 or 2x + 3 = −4
x = 12 or x = −72
Answer: x = 12 or x = −72
No solution, or a coefficient outside and inside.
1|x − 2| + 9 = 4
Isolate the absolute value.
1|x − 2| = −5
Absolute value is never negative.
No solution.
Answer: no solution
1|3x + 1| + 8 = 2
Isolate the absolute value.
1|3x + 1| = −6
Absolute value is never negative.
No solution.
Answer: no solution
4|x − 5| − 7 = 9
Isolate the absolute value.
4|x − 5| = 16
|x − 5| = 4
The formula is
|A| = k means A = k or A = −k.
x − 5 = 4 or x − 5 = −4
x = 9 or x = 1
Answer: x = 9 or x = 1
2|3x − 1| − 8 = 6
Isolate the absolute value.
2|3x − 1| = 14
|3x − 1| = 7
The formula is
|A| = k means A = k or A = −k.
3x − 1 = 7 or 3x − 1 = −7
x = 83 or x = −2
Answer: x = 83 or x = −2
1|5x + 2| − 11 = −3
Isolate the absolute value.
1|5x + 2| = 8
The formula is
|A| = k means A = k or A = −k.
5x + 2 = 8 or 5x + 2 = −8
x = 65 or x = −2
Answer: x = 65 or x = −2
3|2x − 7| + 1 = 1
Isolate the absolute value.
3|2x − 7| = 0
|2x − 7| = 0
2x − 7 = 0
x = 72
Answer: x = 72
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