Question

solve the absolute value equation |4x - 9| = |3x + 2|. (1 point)
no solution
x = -3
x = 11/7 and x = 11
x = 1 and x = 11

Explanation:

Step1: Consider two - case rule for absolute values

If \(|a| = |b|\), then either \(a = b\) or \(a=-b\). So we have two equations: \(4x - 9=3x + 2\) and \(4x - 9=-(3x + 2)\).

Step2: Solve the first equation \(4x - 9=3x + 2\)

Subtract \(3x\) from both sides: \(4x-3x-9=3x - 3x+2\), which simplifies to \(x-9 = 2\). Then add 9 to both sides: \(x=2 + 9\), so \(x = 11\).

Step3: Solve the second equation \(4x - 9=-(3x + 2)\)

Expand the right - hand side: \(4x - 9=-3x - 2\). Add \(3x\) to both sides: \(4x+3x-9=-3x + 3x-2\), which gives \(7x-9=-2\). Add 9 to both sides: \(7x=-2 + 9\), so \(7x = 7\). Divide both sides by 7: \(x = 1\).

Answer:

\(x = 1\) and \(x = 11\)