Question

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 9 right parenthesis Over 3 EndFraction
greater than or equalsStartFraction 3 x Over 7 EndFraction
minus20

Answer 1

To solve the inequality

\[
\frac{5(x - 9)}{3} \geq \frac{3x}{7} - 20,
\]

we'll first eliminate the fractions by finding a common denominator. The common denominator for 3 and 7 is 21.

### Step 1: Multiply both sides by 21
\[
21 \cdot \frac{5(x - 9)}{3} \geq 21 \cdot \left( \frac{3x}{7} - 20 \right).
\]

This gives:
\[
7 \cdot 5(x - 9) \geq 3 \cdot 3x - 420.
\]
\[
35(x - 9) \geq 9x - 420.
\]

### Step 2: Distribute
\[
35x - 315 \geq 9x - 420.
\]

### Step 3: Isolate \( x \)
Subtract \( 9x \) from both sides:
\[
35x - 9x - 315 \geq -420.
\]
\[
26x - 315 \geq -420.
\]

Add 315 to both sides:
\[
26x \geq -105.
\]

### Step 4: Divide by 26
\[
x \geq -\frac{105}{26}.
\]

### Step 5: Simplify the fraction
Calculating the division:
\(-105 ÷ 26 ≈ -4.038...\)

In fraction form, we can leave \( -\frac{105}{26} \) as is.

### Step 6: Interval Notation
The solution in interval notation is
\[
\left[-\frac{105}{26}, \infty\right).
\]

### Graphical Representation
To represent this graphically on a number line:
- Put a solid dot on \( -\frac{105}{26} \) indicating that this point is included in the solution (because it’s ≥).
- Shade the line to the right of this point, indicating all values greater than \( -\frac{105}{26} \).

```
<----•==================>
-105/26
```

Thus, the final answer is:
- Solution: \( x \geq -\frac{105}{26} \)
- Interval Notation: \(\left[-\frac{105}{26}, \infty\right)\)