Question

Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.
StartFraction 3 left parenthesis x minus 6 right parenthesis Over 2 EndFraction
greater than or equalsStartFraction 2 x Over 3 EndFraction
minus12

Answer 1

To solve the inequality

\[
\frac{3(x - 6)}{2} \geq \frac{2x}{3} - 12,
\]

we'll solve it step by step.

### Step 1: Eliminate the fractions

To eliminate the fractions, we can find a common denominator between the fractions. The common denominator between 2 and 3 is 6. We will multiply each term by 6:

\[
6 \cdot \frac{3(x - 6)}{2} \geq 6 \cdot \left(\frac{2x}{3} - 12\right).
\]

This gives us:

\[
9(x - 6) \geq 4x - 72.
\]

### Step 2: Distribute the terms

Distributing \(9\) on the left side:

\[
9x - 54 \geq 4x - 72.
\]

### Step 3: Get all variable terms on one side and constant terms on the other

Subtract \(4x\) from both sides:

\[
9x - 4x - 54 \geq -72.
\]

This simplifies to:

\[
5x - 54 \geq -72.
\]

Now, add \(54\) to both sides:

\[
5x \geq -72 + 54,
\]

which simplifies to:

\[
5x \geq -18.
\]

### Step 4: Solve for \(x\)

Divide both sides by \(5\):

\[
x \geq -\frac{18}{5}.
\]

### Step 5: Interval notation

The solution can be written in interval notation as:

\[
\left[-\frac{18}{5}, \infty\right).
\]

### Step 6: Graphical representation

To represent this on a number line:

1. Draw a number line.
2. Mark the point \(-\frac{18}{5}\) (which is -3.6).
3. Use a closed circle at \(-\frac{18}{5}\) to indicate that this point is included in the solution.
4. Shade the line to the right to indicate that all values greater than \(-\frac{18}{5}\) (including \(-\frac{18}{5}\) itself) are included in the solution.

### Final Answer

Algebraic solution: \(x \geq -\frac{18}{5} \)

Interval notation: \(\left[-\frac{18}{5}, \infty\right)\)

Graphical representation: (close circle at \(-\frac{18}{5}\) and shading to the right)

Here's a rough sketch of the number line:

```
<----(-3.6]================>
```

Where `(-3.6]` indicates the interval starting from \(-\frac{18}{5}\) (closed circle) extending to positive infinity.