How to Solve Inequalities Step by Step

Grade: 7–8 | Topic: Algebra

What You Will Learn

By the end of this page, you will be able to solve one-step and two-step inequalities, represent solutions on a number line, and apply the critical rule about flipping the inequality sign when multiplying or dividing by a negative number.

Theory

Inequality Symbols

Symbol	Meaning	Example
$<$	Less than	$x < 5$
$>$	Greater than	$x > -2$
$\leq$	Less than or equal to	$x \leq 7$
$\geq$	Greater than or equal to	$x \geq 0$

Solving Inequalities Like Equations

You solve inequalities using the same steps as equations — add, subtract, multiply, or divide both sides by the same value.

Critical rule: If you multiply or divide both sides by a negative number, flip the inequality sign.

Why? Because multiplying by $-1$ reverses the number line order: if $3 < 5$, then $-3 > -5$.

Graphing on a Number Line

• Open circle ( ○ ) for $<$ or $>$ — the boundary value is NOT included
• Closed circle ( ● ) for $\leq$ or $\geq$ — the boundary value IS included
• Draw an arrow in the direction of all values that satisfy the inequality

Worked Examples

Example 1: One-Step Inequality (Addition)

Problem: Solve $x + 8 > 3$ and graph the solution.

Step 1: Subtract 8 from both sides. $x + 8 - 8 > 3 - 8$ $x > -5$

Answer: $x > -5$ — open circle at $-5$, arrow pointing right.

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Example 2: One-Step Inequality (Division by Negative)

Problem: Solve $-4x \geq 20$.

Step 1: Divide both sides by $-4$. Flip the sign because we are dividing by a negative. $\frac{-4x}{-4} \leq \frac{20}{-4}$ $x \leq -5$

Answer: $x \leq -5$ — closed circle at $-5$, arrow pointing left.

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Example 3: Two-Step Inequality

Problem: Solve $3x - 4 < 11$.

Step 1: Add 4 to both sides. $3x - 4 + 4 < 11 + 4$ $3x < 15$

Step 2: Divide both sides by 3 (positive — no flip needed). $x < 5$

Answer: $x < 5$ — open circle at 5, arrow pointing left.

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Example 4: Two-Step with Negative Coefficient

Problem: Solve $-2x + 5 \leq 13$.

Step 1: Subtract 5 from both sides. $-2x \leq 8$

Step 2: Divide by $-2$. Flip the sign. $x \geq -4$

Answer: $x \geq -4$ — closed circle at $-4$, arrow pointing right.

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Example 5: Word Problem

Problem: A cinema charges $9 per ticket. Marcus has $45. Write and solve an inequality to find the maximum number of tickets he can buy.

Step 1: Let $t$ = number of tickets. Set up the inequality. $9t \leq 45$

Step 2: Divide both sides by 9. $t \leq 5$

Answer: Marcus can buy at most 5 tickets.

Common Mistakes

Mistake 1: Forgetting to Flip the Sign

❌ $-3x > 12 \implies x > -4$

✅ Dividing by $-3$ flips the sign: $x < -4$

Check: if $x = -5$, then $-3(-5) = 15 > 12$. Correct! If $x = 0$, then $-3(0) = 0$, which is NOT greater than 12. So $x < -4$ is right.

Mistake 2: Flipping When Multiplying by a Positive Number

❌ Student flips the sign when dividing by $+2$.

✅ Only flip when the multiplier or divisor is negative. Positive numbers preserve the direction of the inequality.

Mistake 3: Using the Wrong Circle on the Number Line

❌ $x \leq 4$ graphed with an open circle at 4.

✅ $\leq$ means "less than or equal to" — use a closed circle (the value 4 is a valid solution).

Practice Problems

Try these on your own before checking the answers:

1. Solve $x - 3 \geq 7$
2. Solve $-5x < 25$
3. Solve $2x + 6 > 14$
4. Solve $-3x + 1 \leq 10$
5. A box can hold at most 50 kg. Each bag weighs 3.5 kg. Write and solve an inequality for the number of bags.

Click to see answers

1. $x \geq 10$
2. Divide by $-5$ and flip: $x > -5$
3. $2x > 8 \implies x > 4$
4. $-3x \leq 9$. Divide by $-3$ and flip: $x \geq -3$
5. $3.5b \leq 50 \implies b \leq 14.28...$. So at most 14 bags.

Summary

• Solve inequalities like equations: use inverse operations to isolate the variable.
• Flip the inequality sign whenever you multiply or divide both sides by a negative number.
• Graph: open circle for $<$ or $>$; closed circle for $\leq$ or $\geq$. Arrow shows the direction of valid solutions.
• Check your answer by substituting a value from the solution set back into the original inequality.

Related Topics

• Linear Equations — How to Solve Step by Step
• Solving Two-Step Equations
• Writing and Solving Equations from Word Problems

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