Absolute Value Inequality (One Variable)
How to solve an absolute value inequality (one variable): 2 formulas, 2 examples, and their solutions.
Formula|x| < a
|x| < a
→ -a < x < a.
By the same way,
|x| ≤ a
→ -a ≤ x ≤ a.
Absolute Value
Example|x - 2| < 5
|x - 2| < 5
The left side is an absolute value.
And it is [less than] the right side.
Then, by the above formula,
-5 < x - 2 < 5.
To remove -2 in the middle side,
+2 to each side.
Linear Inequality (One Variable)
-5 + 2 = -3
5 + 2 = 7
So
-3 < x < 7
This is the answer.
Formula|x| > a
|x| < a
→ x < -a or x > a.
By the same way,
|x| ≤ a
→ x ≤ -a or x ≥ a.
Example|2x + 1| ≥ 9
|2x + 1| ≥ 9
The left side is an absolute value.
And it is [greater than] (or equal to) the right side.
Then, by the above formula,
think of two cases.
Case 1: 2x + 1 ≤ -9
Case 2: 2x + 1 ≥ 9
First write case 1:
2x + 1 ≤ -9
Move +1 to the right side.
Then 2x ≤ -10.
Divide both sides by 2.
Then x ≤ -5.
This is the answer for case 1.
Case 2: 2x + 1 ≥ 9
Move +1 to the right side.
Then 2x ≥ 8.
Divide both sides by 2.
Then x ≥ 4.
This is the answer for case 2.
Case 1: x ≤ -5
Case 2: x ≥ 4
Then x ≤ -5 or x ≥ 4.
So
x ≤ -5 or x ≥ 4
is the answer.