Let's solve each inequality step by step and compare their solutions.

First Inequality:

$\frac{1}{2}x + 8 \leq 10$

1. Subtract 8 from both sides: $\frac{1}{2}x \leq 2$
2. Multiply both sides by 2 to eliminate the fraction: $x \leq 4$

Second Inequality:

$-3x - 24 \leq -36$

1. Add 24 to both sides: $-3x \leq -12$
2. Divide both sides by -3 (remember to reverse the inequality sign when dividing by a negative number): $x \geq 4$

Comparing Solutions:

• The first inequality gives $x \leq 4$.
• The second inequality gives $x \geq 4$.
• Together, the only number that satisfies both inequalities is $x = 4$.

Using a Number Line:

A number line helps visualize these solutions.

• The first solution means all values less than or equal to 4 (shaded to the left from 4).
• The second solution means all values greater than or equal to 4 (shaded to the right from 4).
• The overlap is only at $x = 4$, confirming the solution.

Would you like me to provide a number line diagram? Let me know if you have any questions!

5 Related Questions:

1. How do you solve an inequality with fractions or decimals?
2. What happens if you multiply or divide by a negative number in an inequality?
3. How can a number line help compare multiple inequalities?
4. How do you graph solutions of inequalities on a coordinate plane?
5. What is the difference between strict ($<$, $>$) and non-strict ($\leq$, $\geq$) inequalities?

Tip:

When solving inequalities, always check if you need to reverse the inequality sign when multiplying or dividing by a negative number.