Questions: Question 16, 1.6.71 Solve the absolute value equation or indicate that the equation has no solution. 48-(3/2) x+12=40 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Use integers or fractions for any numbers in the expression. Use a comma to separate answers as B. The solution set is the empty set.

Solution Steps

Start by isolating the absolute value expression on one side of the equation. Subtract 12 from both sides: \[ 4\left|8-\frac{3}{2} x\right| + 12 - 12 = 40 - 12 \] Simplify: \[ 4\left|8-\frac{3}{2} x\right| = 28 \]

Divide both sides of the equation by 4 to solve for the absolute value: \[ \left|8-\frac{3}{2} x\right| = \frac{28}{4} \] Simplify: \[ \left|8-\frac{3}{2} x\right| = 7 \]

The absolute value equation \(\left|A\right| = B\) implies two cases: \(A = B\) or \(A = -B\). Apply this to the equation: \[ 8 - \frac{3}{2} x = 7 \quad \text{or} \quad 8 - \frac{3}{2} x = -7 \]

Solve the first case \(8 - \frac{3}{2} x = 7\): \[ 8 - 7 = \frac{3}{2} x \] Simplify: \[ 1 = \frac{3}{2} x \] Multiply both sides by \(\frac{2}{3}\): \[ x = \frac{2}{3} \]

Solve the second case \(8 - \frac{3}{2} x = -7\): \[ 8 + 7 = \frac{3}{2} x \] Simplify: \[ 15 = \frac{3}{2} x \] Multiply both sides by \(\frac{2}{3}\): \[ x = 10 \]

The solutions are \(x = \frac{2}{3}\) and \(x = 10\). Therefore, the solution set is: \[ \boxed{\left\{\frac{2}{3}, 10\right\}} \]

Final Answer