QuestionAugust 11, 2025

Solve 2vert x-1vert =(1)/(2)x+8 for x. no solutions x=-(12)/(5) and x=(20)/(3) x=-(12)/(5) x=(20)/(3)

Solve 2vert x-1vert =(1)/(2)x+8 for x. no solutions x=-(12)/(5) and x=(20)/(3) x=-(12)/(5) x=(20)/(3)
Solve 2vert x-1vert =(1)/(2)x+8 for x. no solutions x=-(12)/(5) and x=(20)/(3) x=-(12)/(5) x=(20)/(3)

Solution
4.2(113 votes)

Answer

x = 6 and x = -\frac{6}{5} Explanation 1. Split into two cases Consider x - 1 \geq 0 and x - 1 < 0 separately. 2. Solve for x - 1 \geq 0 Replace \vert x-1\vert with (x-1): 2(x-1) = \frac{1}{2}x + 8. Simplify to 4x - 2 = x + 16. Solve: 3x = 18, x = 6. Check x - 1 \geq 0: 6 - 1 \geq 0 is true. 3. Solve for x - 1 < 0 Replace \vert x-1\vert with -(x-1): 2(-x+1) = \frac{1}{2}x + 8. Simplify to -2x + 2 = \frac{1}{2}x + 8. Solve: -5x = 6, x = -\frac{6}{5}. Check x - 1 < 0: -\frac{6}{5} - 1 < 0 is true.

Explanation

1. Split into two cases<br /> Consider $x - 1 \geq 0$ and $x - 1 < 0$ separately.<br />2. Solve for $x - 1 \geq 0$<br /> Replace $\vert x-1\vert$ with $(x-1)$: $2(x-1) = \frac{1}{2}x + 8$. Simplify to $4x - 2 = x + 16$. Solve: $3x = 18$, $x = 6$. Check $x - 1 \geq 0$: $6 - 1 \geq 0$ is true.<br />3. Solve for $x - 1 < 0$<br /> Replace $\vert x-1\vert$ with $-(x-1)$: $2(-x+1) = \frac{1}{2}x + 8$. Simplify to $-2x + 2 = \frac{1}{2}x + 8$. Solve: $-5x = 6$, $x = -\frac{6}{5}$. Check $x - 1 < 0$: $-\frac{6}{5} - 1 < 0$ is true.
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