|2x-1| ≥ 7?

1.
|2x-1| ≥ 7 ----------(1)

2x-1 = 7
2x=8
x= 4

2x-1 = -7
2x= -6
x= -3

x=-3; x=4




plot the points on the real line.
------ ------ ------- --------
-∞....-3.....4.....∞

Consider the intervals (-∞,-3),(-3,4),(4,∞)
Choose 'any' one point from each interval and test inequality (1)

(-∞, -3) : choose x=-4
|2x-1| ≥ 7
9 ≥ 7 (true)

(-3,4) : choose x=0
|2x-1| ≥ 7
1 ≥ 7 (false)

(4,∞): choose x=5
|2x-1| ≥ 7
9 ≥ 7 (true)

(-∞,-3] U [4, ∞)

|2x - 1| ≥ 7
|2x - 1|² ≥ 7²
4x² - 4x + 1 ≥ 49
4x² - 4x - 48 ≥ 0
x² - x - 12 ≥ 0
(x + 3)(x - 4) ≥ 0
x ≤ -3 or x ≥ 4

|2x-1| ≥ 7 ---> 2x - 1 ≤ -7, and 2x - 1 ≥ 7.


A) 2x - 1 ≤ -7 ---> 2x ≤ -7 + 1 ---> x ≤ (-7 + 1)/2 ---> x ≤ -6/2 ---> x ≤ -3.

B) 2x - 1 ≥ 7 ---> 2x ≥ 7 + 1 ---> x ≥ (7 + 1)/2 ---> x ≥ 8/2 ---> x ≥ 4.


Answer: x ≤ -3; x ≥ 4.

|a|= √(a²) so |2x-1| ≥ 7
→ √((2x-1)²) ≥ 7
→ (2x-1)² ≥ 49
→ 4x²-4x+1 ≥ 49
→ 4(x²-x-12) ≥ 0
→ 4(x+3)(x-4) ≥ 0
→ x ≤ -3 or x ≥ 4

Graph comparing |2x-1| ≥ 7 and 4(x+3)(x-4) ≥ 0: https://www.desmos.com/calculator/glybc4lypc

|2x - 1| >= 7

2x - 1 >= 7 or 2x - 1 <= -7

2x >= 8 or 2x <= -6

x >= 4 or x <= -3 (interval notation: [-oo, -3] U [4, oo]).

2x - 1 ≥ 7

2x ≥ 8

x ≥ 4

2x - 1 ≦ - 7

2x ≦ - 6

x ≦ - 3

Solution ( - inf, - 3] U [ 4, + inf)

Are you supposed to graph it?

|2x - 1| ≥ 7

An absolute value is always ≥ 0, so your inequality is always true whatever the value of x.

→ Solution = IR

4 <= x <= -3

2x - 1 ≤ - 7___or ____ 7 ≤ 2x - 1
2x ≤ -6_____ or_____2x ≥ - 8
x ≤ - 3_____or_______x ≥ - 4

solution set is { - 4 ≤ x ≤ - 3 }

2x-1 ≥ 7 or 2x-1 ≤ -7
2x ≥ 8 or 2x ≤ -6
x ≥ 4 or x ≤ -3
(-∞,-3) U (4,∞)