22. $$ -3\left|1-\frac{2}{3} v\right|=-9 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
-3\left|1 - \frac{2}{3}v\right| = -9
$$

**Step 1: Isolate the Absolute Value**

Divide both sides by $$-3$$ to isolate the absolute value:

$$
\left|1 - \frac{2}{3}v\right| = \frac{-9}{-3} \implies \left|1 - \frac{2}{3}v\right| = 3
$$

**Step 2: Remove the Absolute Value**

An equation of the form $$|A| = B$$ has two solutions: $$A = B$$ or $$A = -B$$. Apply this to the equation:

1. **First Case:**
   $$
   1 - \frac{2}{3}v = 3
   $$
   Subtract 1 from both sides:
   $$
   -\frac{2}{3}v = 2
   $$
   Multiply both sides by $$-\frac{3}{2}$$:
   $$
   v = -3
   $$

2. **Second Case:**
   $$
   1 - \frac{2}{3}v = -3
   $$
   Subtract 1 from both sides:
   $$
   -\frac{2}{3}v = -4
   $$
   Multiply both sides by $$-\frac{3}{2}$$:
   $$
   v = 6
   $$

**Solution:**

The values of $$v$$ that satisfy the equation are:

$$
v = -3 \quad \text{and} \quad v = 6
$$

**Verification:**

To ensure these solutions are correct, substitute them back into the original equation.

1. **For $$v = -3$$:**
   $$
   -3\left|1 - \frac{2}{3}(-3)\right| = -3\left|1 + 2\right| = -3(3) = -9
   $$
   Which matches the right side.

2. **For $$v = 6$$:**
   $$
   -3\left|1 - \frac{2}{3}(6)\right| = -3\left|1 - 4\right| = -3(3) = -9
   $$
   Which also matches the right side.

Both solutions are valid.

**Final Answer:**
$$ v = -3 \quad \text{and} \quad v = 6 $$
The solutions are $$ v = -3 $$ and $$ v = 6 $$.

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Beyond the Answer

For the First Case:

For the Second Case:

Final Solutions:

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