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: \(\left[-\frac{105}{26}, \infty\right)\)
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