Problem 86 - Algebra and Trigonometry

Question

Find all values of \(x\) satisfying the given conditions. $$y=|2-3 x| \text { and } y=13$$

Step-by-Step Solution

Answer

The solutions to the equations are \(x = - \frac{11}{3}\) and \(x = 5\).

Step 1: Isolate the Absolute Value Expression

The absolute value equation given is \(y=|2-3 x|\). This needs to be isolated to look only at the absolute value equation: \(|2-3x| = y\).

Step 2: Substitute y Value

According to the conditions, \(y=13\), substitute \(y = 13\) into the equation, then the equation will now become \(|2-3x|=13\)

Step 3: Create Two Equations

Since absolute value represents distance and can either be positive or negative, we divide into two equations: \(2-3x=13\) and \(2-3x=-13\).

Step 4: Solve for x in Both Equations

Start to solve for each equation. \n\nFor \(2-3x=13\):\n- Subtract 2 from both sides: -3x = 11\n- Divide by -3: \(x = - \frac{11}{3}\)\n\nFor \(2-3x=-13\):\n- Subtract 2 from both sides: -3x = -15\n- Divide by -3: \(x = 5\)