Solving Inequalities with variables on both sides of the Sign ... - Free Printable

Let's solve each inequality step by step. We'll simplify, isolate the variable, and remember that if we multiply or divide by a negative number, we must reverse the inequality sign.

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1. $-3(x + 5) < 3(x - 1)$

Step 1: Distribute both sides
$$
-3x - 15 < 3x - 3
$$

Step 2: Move all terms with $x$ to one side and constants to the other
Add $3x$ to both sides:
$$
-15 < 6x - 3
$$

Add 3 to both sides:
$$
-12 < 6x
$$

Step 3: Divide by 6
$$
-2 < x \quad \text{or} \quad x > -2
$$

✔ Answer: $x > -2$

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2. $5(x + 2) \leq 2(3 - x)$

Step 1: Distribute
$$
5x + 10 \leq 6 - 2x
$$

Step 2: Move variables to one side
Add $2x$ to both sides:
$$
7x + 10 \leq 6
$$

Subtract 10:
$$
7x \leq -4
$$

Step 3: Divide by 7
$$
x \leq -\frac{4}{7}
$$

✔ Answer: $x \leq -\frac{4}{7}$

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3. $4(m + 3) > 36$

Step 1: Distribute
$$
4m + 12 > 36
$$

Step 2: Subtract 12
$$
4m > 24
$$

Step 3: Divide by 4
$$
m > 6
$$

✔ Answer: $m > 6$

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4. $4(y - 2) < 2(5 - y)$

Step 1: Distribute
$$
4y - 8 < 10 - 2y
$$

Step 2: Move variables to one side
Add $2y$ to both sides:
$$
6y - 8 < 10
$$

Add 8:
$$
6y < 18
$$

Step 3: Divide by 6
$$
y < 3
$$

✔ Answer: $y < 3$

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5. $3(x - 8) \geq 3x$

Step 1: Distribute
$$
3x - 24 \geq 3x
$$

Step 2: Subtract $3x$ from both sides
$$
-24 \geq 0
$$

✘ This is false — it's never true.

So there is no solution.

✔ Answer: No solution (or $\emptyset$)

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✔ Final Answers:

1. $x > -2$
2. $x \leq -\frac{4}{7}$
3. $m > 6$
4. $y < 3$
5. No solution

Let me know if you'd like these graphed on a number line!