Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.
StartFraction 5 left parenthesis x minus 8 right parenthesis Over 4 EndFraction
greater than or equalsStartFraction 4 x Over 5 EndFraction
minus15
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Part 1
The solution as an inequality is

### Part 1: Solving the inequality algebraically

We are given the inequality:

\[
\frac{5(x - 8)}{4} \geq \frac{4x}{5} - 15
\]

#### Step 1: Eliminate the denominators

Multiply both sides of the inequality by the least common denominator (LCD) of 4 and 5, which is 20:

\[
20 \cdot \frac{5(x - 8)}{4} \geq 20 \cdot \left(\frac{4x}{5} - 15\right)
\]

Simplifying each side:

\[
5 \cdot 5(x - 8) \geq 4 \cdot 4x - 300
\]

\[
25(x - 8) \geq 16x - 300
\]

#### Step 2: Expand both sides

Expand both expressions:

\[
25x - 200 \geq 16x - 300
\]

#### Step 3: Collect like terms

Move the \(16x\) to the left side and the constants to the right side:

\[
25x - 16x \geq -300 + 200
\]

\[
9x \geq -100
\]

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

Divide both sides by 9:

\[
x \geq \frac{-100}{9}
\]

So the solution as an inequality is:

\[
x \geq \frac{-100}{9}
\]

### Part 2: Graphing the solution on a number line

On a number line, the solution represents all values of \(x\) that are greater than or equal to \(-\frac{100}{9}\). This is approximately \(-11.11\), so the number line would show a closed circle at \(-\frac{100}{9}\) with an arrow pointing to the right (indicating all greater values).

### Part 3: Interval notation

The solution in interval notation is:

\[
\left[\frac{-100}{9}, \infty \right)
\]

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Would you like me to generate the graphical number line representation?

#### Follow-up Questions:
1. What are the steps involved in solving inequalities involving fractions?
2. How do you interpret the solution of an inequality in terms of a graph?
3. How can you verify the solution of this inequality?
4. What role does the least common denominator play in solving inequalities?
5. How does multiplying or dividing by a negative number affect an inequality?

**Tip**: When solving inequalities, be mindful of flipping the inequality sign when multiplying or dividing by a negative number!

Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation. \( \frac{5(x - 8)}{4} \geq \frac{4x}{5} - 15 \)

Explore how to solve the inequality \( \frac{5(x - 8)}{4} \geq \frac{4x}{5} - 15 \) algebraically and graphically. This problem covers key algebra concepts, including eliminating fractions, using the least common denominator, and representing the solution on a number line. Suitable for high school students in Grades 9-12.

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