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Question Solve the absolute value equation ∣frac6x+33∣+4=8|\\frac{6 x+3}{3}|+4=8 and express the solution set.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation ∣6x+33∣+4=8\left|\frac{6x+3}{3}\right|+4=8.
2. We need to solve for the variable xx.
3. The absolute value function ∣a∣|a| is defined as aa if a≥0a \geq 0 and −a-a if a<0a < 0.

STEP 2

First, we will isolate the absolute value expression by subtracting 4 from both sides of the equation.
∣6x+33∣+4−4=8−4\left|\frac{6x+3}{3}\right|+4-4=8-4

STEP 3

Simplify the equation.
∣6x+33∣=4\left|\frac{6x+3}{3}\right|=4

STEP 4

Now we will set up two separate equations to account for the definition of the absolute value. The first equation is for the case where the expression inside the absolute value is non-negative, and the second equation is for when the expression inside is negative.
6x+33=4and6x+33=−4\frac{6x+3}{3}=4 \quad \text{and} \quad \frac{6x+3}{3}=-4

STEP 5

Solve the first equation where the expression inside the absolute value is non-negative.
6x+33=4\frac{6x+3}{3}=4

STEP 6

Multiply both sides by 3 to eliminate the denominator.
6x+3=4×36x+3=4 \times 3

STEP 7

Simplify the right side of the equation.
6x+3=126x+3=12

STEP 8

Subtract 3 from both sides to isolate the term with xx.
6x+3−3=12−36x+3-3=12-3

STEP 9

Simplify the equation.
6x=96x=9

STEP 10

Divide both sides by 6 to solve for xx.
x=96x=\frac{9}{6}

STEP 11

Simplify the fraction.
x=32x=\frac{3}{2}

STEP 12

Now, solve the second equation where the expression inside the absolute value is negative.
6x+33=−4\frac{6x+3}{3}=-4

STEP 13

Multiply both sides by 3 to eliminate the denominator.
6x+3=−4×36x+3=-4 \times 3

STEP 14

Simplify the right side of the equation.
6x+3=−126x+3=-12

STEP 15

Subtract 3 from both sides to isolate the term with xx.
6x+3−3=−12−36x+3-3=-12-3

STEP 16

Simplify the equation.
6x=−156x=-15

STEP 17

Divide both sides by 6 to solve for xx.
x=−156x=\frac{-15}{6}

STEP 18

Simplify the fraction.
x=−52x=-\frac{5}{2}

STEP 19

Combine the solutions from STEP_11 and STEP_18 into a solution set using set notation.
Solution set={−52,32}\text{Solution set} = \left\{ -\frac{5}{2}, \frac{3}{2} \right\}
The solution set is {−52,32}\left\{ -\frac{5}{2}, \frac{3}{2} \right\}.

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