Question Solve the equation $|3x+2| - 8 = 2x$ for all values of $x$ .

Studdy Solution

STEP 1

Assumptions
1. The equation to solve is $|3x + 2| - 8 = 2x$ .
2. The solution involves considering two cases because the expression inside the absolute value can be either positive or negative.

STEP 2

Understand the definition of absolute value. The absolute value of a number is its distance from zero on the number line, without considering direction. For any real number $a$ , $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$ .

STEP 3

Set up the first case for when the expression inside the absolute value is non-negative, which means $3x + 2 \geq 0$ .

STEP 4

Solve the equation for the first case:
$|3x + 2| - 8 = 2x$
becomes
$3x + 2 - 8 = 2x$

STEP 5

Subtract $2x$ from both sides to isolate $x$ on one side:
$3x - 2x + 2 - 8 = 0$

STEP 6

Combine like terms:
$x - 6 = 0$

STEP 7

Add 6 to both sides to solve for $x$ :
$x = 6$

STEP 8

Check if the solution $x = 6$ satisfies the original condition $3x + 2 \geq 0$ :
$3(6) + 2 \geq 0$
$18 + 2 \geq 0$
$20 \geq 0$
which is true, so $x = 6$ is a valid solution for the first case.

STEP 9

Set up the second case for when the expression inside the absolute value is negative, which means $3x + 2 < 0$ .

STEP 10

Solve the equation for the second case:
$|3x + 2| - 8 = 2x$
becomes
$-(3x + 2) - 8 = 2x$

STEP 11

Distribute the negative sign:
$-3x - 2 - 8 = 2x$

STEP 12

Add $3x$ to both sides to start gathering $x$ terms on one side:
$-2 - 8 = 2x + 3x$

STEP 13

Combine like terms:
$-10 = 5x$

STEP 14

Divide both sides by 5 to solve for $x$ :
$x = -\frac{10}{5}$

STEP 15

Simplify the fraction:
$x = -2$

STEP 16

Check if the solution $x = -2$ satisfies the original condition $3x + 2 < 0$ :
$3(-2) + 2 < 0$
$-6 + 2 < 0$
$-4 < 0$
which is true, so $x = -2$ is a valid solution for the second case.

STEP 17

Combine the solutions from both cases. The solutions to the equation $|3x + 2| - 8 = 2x$ are:
$x = 6 \quad \text{and} \quad x = -2$

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