Inequalities Part 3 Quiz

|2x − 3| ≥ 7
Case 1:
2x − 3 ≥ 7
x ≥ 5
Case 2:
2x − 3 ≤ −7
x ≤ −2
Solution: x ≤ −2 or x ≥ 5

−5 < 2x + 1 ≤ 7 or, −5 − 1 < 2x + 1 − 1 ≤ 7 − 1 or, −6 < 2x ≤ 6 or, −3 < x ≤ 3

(x − 4)(x + 2) > 0
or, Critical points: x = 4 and x = −2
So the solution is x < −2 or x > 4

|x| − 3 < 1 or, |x| < 4 Solution: −4 < x < 4

(2x + 5)/(x − 1) ≤ 0
Critical points:
x = 1 (denominator = 0, undefined)
x = −5/2 (numerator = 0)
−5/2 ≤ x < 1

|3x − 5| < |x + 1| Square both and solve for the quadratic eq 9x² − 30x + 25 < x² + 2x + 1 (x − 1)(x − 3) < 0 Solution: 1 < x < 3

(x² − 4)/(x − 2) > 0
(x − 2)(x + 2) / (x − 2) > 0
Critical points: x = 2 (denominator = 0, undefined), x = −2 (numerator = 0)
Solx > −2, x ≠ 2

|x − 1| + 2 ≥ 5
or, |x − 1| ≥ 3
Case 1: x − 1 ≥ 3 → x ≥ 4
Case 2: x − 1 ≤ −3 → x ≤ −2
Solution: x ≤ −2 or x ≥ 4

2x + 3 ≤ x²
or, x² − 2x − 3 ≥ 0
or, (x − 3)(x + 1) ≥ 0
Critical points: x = 3, x = −1
Solution: x ≤ −1 or x ≥ 3

|x² − 9| = 9
Case 1:
x² − 9 = 9
x = ±√18
Case 2:
x² − 9 = −9
x = 0
Solution: x = ±√18, x = 0