Solve the inequality x/4 + 2 ≥ 4 and represent the solution on a number line.
Understand the Problem
The question is asking to solve the inequality ( \frac{x}{4} + 2 \geq 4 ) and to provide the solution on a number line.
Answer
The solution to the inequality is \( x \geq 8 \).
Answer for screen readers
The solution to the inequality is ( x \geq 8 ).
Steps to Solve
- Isolate the variable term
Subtract 2 from both sides of the inequality:
$$ \frac{x}{4} + 2 - 2 \geq 4 - 2 $$
This simplifies to:
$$ \frac{x}{4} \geq 2 $$
- Remove the fraction
Multiply both sides by 4 to eliminate the fraction (since 4 is a positive number, the direction of the inequality remains the same):
$$ 4 \cdot \frac{x}{4} \geq 4 \cdot 2 $$
This simplifies to:
$$ x \geq 8 $$
- Draw the number line
On the number line, indicate the point where ( x = 8 ) and shade all values to the right of 8 to represent all numbers greater than or equal to 8. The point at 8 should be solid, indicating that it is included in the solution set.
The solution to the inequality is ( x \geq 8 ).
More Information
This inequality means that any number equal to or greater than 8 satisfies the original inequality. When graphed on a number line, it indicates not only the point at 8 but all numbers to the right.
Tips
- Forgetting to flip the inequality when multiplying or dividing by a negative number (not applicable here as we only multiplied by a positive number).
- Misinterpreting the solid and open dots on the number line; remember that a solid dot indicates inclusion (equal to) while an open dot would indicate exclusion.
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