To solve the equation [tex]\(\left|\frac{2}{3} q - 1\right| - 1 = 4\)[/tex], we can follow these steps:
1. Isolate the Absolute Value:
[tex]\[
\left|\frac{2}{3} q - 1\right| - 1 = 4
\][/tex]
Add 1 to both sides:
[tex]\[
\left|\frac{2}{3} q - 1\right| = 5
\][/tex]
2. Set Up Two Separate Equations:
The equation [tex]\(\left| A \right| = B\)[/tex] yields two equations:
[tex]\[
A = B \quad \text{and} \quad A = -B
\][/tex]
In our case, [tex]\(A = \frac{2}{3} q - 1\)[/tex] and [tex]\(B = 5\)[/tex]. So, we write:
[tex]\[
\frac{2}{3} q - 1 = 5
\][/tex]
and
[tex]\[
\frac{2}{3} q - 1 = -5
\][/tex]
3. Solve Each Equation:
- For [tex]\(\frac{2}{3} q - 1 = 5\)[/tex]:
[tex]\[
\frac{2}{3} q - 1 = 5
\][/tex]
Add 1 to both sides:
[tex]\[
\frac{2}{3} q = 6
\][/tex]
Multiply both sides by [tex]\(\frac{3}{2}\)[/tex] to solve for [tex]\(q\)[/tex]:
[tex]\[
q = 6 \times \frac{3}{2} = 9
\][/tex]
- For [tex]\(\frac{2}{3} q - 1 = -5\)[/tex]:
[tex]\[
\frac{2}{3} q - 1 = -5
\][/tex]
Add 1 to both sides:
[tex]\[
\frac{2}{3} q = -4
\][/tex]
Multiply both sides by [tex]\(\frac{3}{2}\)[/tex] to solve for [tex]\(q\)[/tex]:
[tex]\[
q = -4 \times \frac{3}{2} = -6
\][/tex]
4. Write the Final Solution:
The solutions to the equation [tex]\(\left|\frac{2}{3} q - 1\right| - 1 = 4\)[/tex] are:
[tex]\[
\boxed{q = 9 \quad \text{and} \quad q = -6}
\][/tex]
Hence, [tex]\(q = 9\)[/tex] and [tex]\(q = -6\)[/tex] are the values that satisfy the given equation.
