Problem: Solve the inequality 2x − 5 < 9 and graph its solution on a number line.
Step 1: Add 5 to both sides to begin isolating x.
2x−5+5<9+5⟹2x<14
Step 2: Divide both sides by 2. Since 2 is positive, the inequality direction stays the same.
22x<214⟹x<7
Step 3: Interpret the solution. Every number less than 7 satisfies the inequality. On a number line, draw an open circle at 7 (because 7 itself is not included) and shade everything to the left.
Step 4: Verify with a test value. Choose x = 3, which is less than 7.
2(3)−5=1<9✓
Answer: The solution set is x < 7, or in interval notation (−∞, 7).
Another Example
Problem: Solve the inequality −3x + 6 ≥ 12.
Step 1: Subtract 6 from both sides.
−3x+6−6≥12−6⟹−3x≥6
Step 2: Divide both sides by −3. Because you are dividing by a negative number, you must flip the inequality sign.
−3−3x≤−36⟹x≤−2
Step 3: Verify with a test value. Choose x = −4, which is less than −2.
−3(−4)+6=12+6=18≥12✓
Answer: The solution set is x ≤ −2, or in interval notation (−∞, −2].
Frequently Asked Questions
What is the difference between an equation and an inequality?
An equation uses the = sign and states that two expressions have the same value, typically producing a finite number of solutions. An inequality uses <, >, ≤, or ≥ and states that one expression is larger or smaller than the other, usually producing a range (infinitely many values) as its solution set.
Why do you flip the inequality sign when multiplying or dividing by a negative number?
Multiplying or dividing by a negative number reverses the order of numbers on the number line. For example, 2 < 5 is true, but if you multiply both sides by −1 you get −2 and −5. Since −2 > −5, the sign must flip to keep the statement true.
Equation vs. Inequality
An equation (e.g., 2x + 1 = 7) asserts that two expressions are equal and usually has one or a few specific solutions. An inequality (e.g., 2x + 1 < 7) asserts that one expression is greater or less than the other and typically has a whole range of solutions. You solve both using similar algebraic steps, but with inequalities you must reverse the sign whenever you multiply or divide by a negative number.
Why It Matters
Inequalities model real-world constraints such as budgets, speed limits, and minimum requirements—situations where values must stay above or below a threshold rather than hit an exact target. They are foundational in algebra, calculus, and optimization, where you need to describe entire regions of valid solutions rather than single points. Nearly every field that uses mathematics—engineering, economics, computer science—relies on inequalities to define feasible conditions.
Common Mistakes
Mistake: Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number.
Correction: Whenever you multiply or divide both sides of an inequality by a negative value, reverse the direction of the inequality symbol. For example, dividing −2x > 8 by −2 gives x < −4, not x > −4.
Mistake: Using an open circle on a number line when the inequality includes 'or equal to' (≤ or ≥).
Correction: Use a closed (filled-in) circle for ≤ or ≥ because the endpoint is part of the solution. Use an open circle only for strict inequalities (< or >).
Related Terms
Expression — The building blocks compared in an inequality
