Solve the absolute value equation or indicate that the equation has no solution.
-$$\left| \frac { 1 } { 2 } x + 2 \right| = \left| \frac { 3 } { 4 } x - 2 \right|$$
A) $\{ 16,0 \}$
B) $\{ 16,12 \}$
C) $\{ 10,10 \}$
D) $\varnothing$

Question

Quizplus · Accepted Answer

To solve the equation $\left| \frac { 1 } { 2 } x + 2 \right| = \left| \frac { 3 } { 4 } x - 2 \right|$, we consider two cases due to the properties of absolute values:1. $\frac { 1 } { 2 } x + 2 = \frac { 3 } { 4 } x - 2$Solving for $x$, we subtract $\frac { 1 } { 2 } x$ from both sides and add 2 to both sides, getting $4 = \frac { 1 } { 4 } x$, which gives $x = 16$.2. $\frac { 1 } { 2 } x + 2 = -(\frac { 3 } { 4 } x - 2)$Solving for $x$, we have $\frac { 1 } { 2 } x + 2 = -\frac { 3 } { 4 } x + 2$. Adding $\frac { 3 } { 4 } x$ to both sides and subtracting 2 from both sides, we get $\frac { 5 } { 4 } x = 0$, which gives $x = 0$.Therefore, the solution set is $\{ 16,0 \}$.

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Solve the Absolute Value Equation or Indicate That the Equation $\left| \frac { 1 } { 2 } x + 2 \right| = \left| \frac { 3 } { 4 } x - 2 \right|$