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:
- Draw a number line.
- Mark the point \(-\frac{18}{5}\) (which is -3.6).
- Use a closed circle at \(-\frac{18}{5}\) to indicate that this point is included in the solution.
- 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.
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