Inequalities

An inequality is like an equation, except instead of an equals sign it uses a comparison symbol. The solution is not a single number — it is a range of values that make the statement true.

Inequality Symbols

The difference between $<$ and $\leq$ matters. With $x < 5$, the number 5 itself is not included. With $x \leq 5$, the number 5 is included.

Solving One-Variable Inequalities

Solving inequalities follows the same steps as solving equations with one critical exception:

When you multiply or divide both sides by a negative number, you must flip the inequality sign.

This is the single most important rule in this topic. It trips up students more than anything else.

Why the Sign Flips

Consider the true statement $2 < 6$. Multiply both sides by $-1$:

$-2 \quad ? \quad -6$

Since $-2$ is to the right of $-6$ on the number line, $-2 > -6$. The inequality flipped from $<$ to $>$.

Example 1: One-Step Inequality

Solve $x + 4 > 9$.

Subtract 4 from both sides:

$x > 5$

Answer: $x > 5$. Any number greater than 5 is a solution.

Example 2: Two-Step Inequality

Solve $3x - 7 \leq 11$.

Step 1 — Add 7 to both sides:

$3x \leq 18$

Step 2 — Divide both sides by 3:

$x \leq 6$

Answer: $x \leq 6$. Any number less than or equal to 6 is a solution.

Example 3: Dividing by a Negative (Sign Flip)

Solve $-4x + 2 > 14$.

Step 1 — Subtract 2 from both sides:

$-4x > 12$

Step 2 — Divide both sides by $-4$ and flip the sign:

$x < -3$

Answer: $x < -3$. Dividing by $-4$ flipped $>$ to $<$.

Check: Try $x = -5$ (which is less than $-3$): $-4(-5) + 2 = 20 + 2 = 22$. Is $22 > 14$? Yes. The solution checks out.

Graphing Inequalities on a Number Line

Representing solutions visually helps you see the range of answers:

• Open circle ($\circ$) at the boundary for $<$ or $>$ — the boundary is not included
• Closed circle ($\bullet$) at the boundary for $\leq$ or $\geq$ — the boundary is included
• Arrow extends in the direction of the solution

For $x > 5$: open circle at 5, arrow pointing right

For $x \leq 6$: closed circle at 6, arrow pointing left

For $x < -3$: open circle at $-3$, arrow pointing left

Compound Inequalities

A compound inequality combines two inequalities. There are two types:

“And” Compound Inequalities

Both conditions must be true at the same time. Often written in combined form:

$a < x < b \quad \text{means} \quad x > a \;\text{ AND }\; x < b$

Example 4: Solve $-2 \leq 3x + 1 < 10$

Work with all three parts simultaneously. Subtract 1 from all three parts:

$-2 - 1 \leq 3x < 10 - 1$

$-3 \leq 3x < 9$

Divide all three parts by 3:

$-1 \leq x < 3$

Answer: $x$ is between $-1$ (inclusive) and $3$ (exclusive). On a number line: closed circle at $-1$, open circle at $3$, shading in between.

”Or” Compound Inequalities

At least one condition must be true. The solution is the union of both solution sets.

Example 5: Solve $2x + 1 < -3$ or $2x + 1 > 7$

Solve each separately:

First inequality: $2x < -4 \implies x < -2$

Second inequality: $2x > 6 \implies x > 3$

Answer: $x < -2$ or $x > 3$. The solution is everything to the left of $-2$ and everything to the right of 3, with a gap in the middle.

Real-World Application: Nursing — Safe Dosage Ranges

A nurse is administering a medication where the safe dosage range is based on the patient’s weight. For a patient weighing 70 kg, the medication guidelines state:

• Minimum dose: $5$ mg per kg
• Maximum dose: $8$ mg per kg

The safe total dosage $D$ (in mg) must satisfy:

$5 \times 70 \leq D \leq 8 \times 70$

$350 \leq D \leq 560$

The medication comes in a concentration of 25 mg/mL. The safe volume $V$ (in mL) to administer is:

$\frac{350}{25} \leq V \leq \frac{560}{25}$

$14 \leq V \leq 22.4$

Answer: The nurse should administer between 14 mL and 22.4 mL of the medication. Administering less than 14 mL would be subtherapeutic (not enough to be effective), and more than 22.4 mL could cause adverse effects. The compound inequality defines the safe window precisely.

Common Mistakes to Avoid

1. Forgetting to flip the inequality sign. This is the number-one mistake. Any time you multiply or divide by a negative number, the sign reverses.

   $-2x > 6 \implies x < -3 \quad \checkmark$ $-2x > 6 \implies x > -3 \quad \text{(wrong — forgot to flip)}$

2. Flipping the sign when adding or subtracting a negative. The flip rule applies only to multiplication and division, not to addition and subtraction. Subtracting a number from both sides does not flip the sign.

3. Confusing open and closed circles. Strict inequalities ($<$, $>$) use open circles. Non-strict inequalities ($\leq$, $\geq$) use closed circles.

4. Reading compound inequalities in the wrong direction. The statement $-1 \leq x < 3$ means $x$ is at least $-1$ and less than $3$. Make sure the variable is in the middle.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Inequalities are solved just like equations, except you must flip the inequality sign when multiplying or dividing by a negative number
• Solutions are ranges, not single numbers — graph them on a number line with open or closed circles
• Open circles for strict ($<$, $>$); closed circles for non-strict ($\leq$, $\geq$)
• Compound inequalities combine two conditions: “and” means both must be true (intersection), “or” means at least one must be true (union)
• In real-world applications, inequalities define safe ranges, budgets, tolerances, and limits

Return to Algebra for more topics in this section.