Inequalities | FREE ACT Math Course | The School of Mathematics

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<h1>ACT Math: Inequalities</h1>

Inequalities extend equation-solving to situations with a range of possible answers, and they're tested through solving, graphing, and word problems alike. This free, complete lesson covers solving and graphing linear inequalities, including the crucial sign-flip rule, compound inequalities, absolute value inequalities, and real-world word problems that translate into inequalities. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-inequalities-quiz-1">Practice Inequalities Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#linear">Solving and Graphing Linear Inequalities</a></li>

<li><a href="#compound">Compound Inequalities</a></li>

<li><a href="#absolute-value">Absolute Value Inequalities</a></li>

<li><a href="#word-problems">Word Problems and Applications</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

</nav>

<h2 id="linear">1. Solving and Graphing Linear Inequalities</h2>

<p>Solving an inequality works just like solving an equation, with one critical exception: <strong>multiplying or dividing both sides by a negative number reverses the inequality symbol.</strong></p>

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<p class="figure-caption">x ≥ −5: closed circle, shaded to the right</p>

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<p><strong>Worked Example:</strong> Solve −3x + 7 ≤ 22.</p>

<p>Divide both sides by −3, and flip the inequality symbol:</p>

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<p class="prompt">1. Solve: x − 5 > 12</p>

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<p><span class="label">Explanation:</span> Add 5 to both sides: x > 17.</p>

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<p class="prompt">2. Solve: 3x ≤ 21</p>

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<p><span class="label">Explanation:</span> Divide both sides by 3 (positive, so the symbol doesn't flip): x ≤ 7.</p>

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<p class="prompt">3. Solve: −4x < 20</p>

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<p><span class="label">Explanation:</span> Divide both sides by −4 (negative, so flip the symbol): x > −5.</p>

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<p class="prompt">4. Solve: 2x + 5 ≥ 17</p>

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<p><span class="label">Explanation:</span> Subtract 5: 2x ≥ 12. Divide by 2: x ≥ 6.</p>

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<p class="prompt">5. Solve: −3x + 7 ≤ 22</p>

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<p><span class="label">Explanation:</span> This is the worked example above: subtract 7, then divide by −3, flipping the symbol, giving x ≥ −5.</p>

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<h2 id="compound">2. Compound Inequalities</h2>

<p>A <strong>compound inequality</strong> like a < expression ≤ b combines two conditions at once. Solve it by performing the same operation to all three parts simultaneously.</p>

<p><strong>Worked Example:</strong> Solve −3 < 2x + 1 ≤ 9.</p>

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<p class="prompt">1. Solve: −6 ≤ 3x < 12</p>

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<p><span class="label">Explanation:</span> Divide all three parts by 3: −2 ≤ x < 4.</p>

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<p class="prompt">2. Solve: 5 < x + 2 < 9</p>

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<p><span class="label">Explanation:</span> Subtract 2 from all three parts: 3 < x < 7.</p>

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<p class="prompt">3. Solve: −2 ≤ <span class="frac"><span class="num">x − 3</span><span class="den">2</span></span> ≤ 5</p>

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<p><span class="label">Explanation:</span> Multiply all three parts by 2: −4 ≤ x − 3 ≤ 10. Add 3 to all three parts: −1 ≤ x ≤ 13.</p>

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<p class="prompt">4. Which compound inequality represents all real numbers greater than −2 AND less than or equal to 6?</p>

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<p><span class="label">Explanation:</span> "Greater than −2" excludes −2, using <. "Less than or equal to 6" includes 6, using ≤. Together: −2 < x ≤ 6.</p>

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<p class="prompt">5. Solve: −12 < 4x ≤ 24</p>

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<p><span class="label">Explanation:</span> Divide all three parts by 4: −3 < x ≤ 6.</p>

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<h2 id="absolute-value">3. Absolute Value Inequalities</h2>

<p>An absolute value inequality splits into two cases. For |expression| < k (a "less than" inequality), the result is a single compound inequality: −k < expression < k. For |expression| > k (a "greater than" inequality), the result is two separate, non-overlapping inequalities: expression > k, or expression < −k.</p>

<p><strong>Worked Example:</strong> Solve |x − 4| < 6.</p>

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<p class="prompt">1. Solve: |x + 2| < 7</p>

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<p><span class="label">Explanation:</span> −7 < x + 2 < 7. Subtract 2 from all parts: −9 < x < 5.</p>

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<p class="prompt">2. Solve: |3x − 6| ≤ 12</p>

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<p><span class="label">Explanation:</span> −12 ≤ 3x − 6 ≤ 12. Add 6: −6 ≤ 3x ≤ 18. Divide by 3: −2 ≤ x ≤ 6.</p>

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<p class="prompt">3. Solve: |x − 5| > 3</p>

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<p><span class="label">Explanation:</span> Since this is a "greater than," split into two separate cases: x − 5 > 3 gives x > 8, and x − 5 < −3 gives x < 2.</p>

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<p class="prompt">4. Solve: |2x + 1| ≥ 9</p>

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<p><span class="label">Explanation:</span> 2x + 1 ≥ 9 gives x ≥ 4. 2x + 1 ≤ −9 gives x ≤ −5.</p>

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<p class="prompt">5. Which inequality describes all values of x within 4 units of 10?</p>

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<p><span class="label">Explanation:</span> "Within 4 units of 10" means the distance between x and 10 is less than 4, so |x − 10| < 4.</p>

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<h2 id="word-problems">4. Word Problems and Applications</h2>

<p>Inequality word problems often involve phrases like "at least" (≥), "at most" (≤), "no more than" (≤), "exceeds" (>), or "within" (an absolute value inequality). Translate the phrase carefully before setting up the inequality.</p>

<p class="prompt">1. A parking garage charges $5 for the first hour and $3 for each additional hour. If Sam wants to pay no more than $23 total, what is the maximum number of additional hours he can park?</p>

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<p><span class="label">Explanation:</span> Let h = additional hours: 5 + 3h ≤ 23. Subtract 5: 3h ≤ 18. Divide by 3: h ≤ 6. The maximum is 6 additional hours.</p>

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<p class="prompt">2. A number tripled and then increased by 4 is at least 25. What is the smallest possible integer value of the number?</p>

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<p><span class="label">Explanation:</span> Let x = the number: 3x + 4 ≥ 25. Subtract 4: 3x ≥ 21. Divide by 3: x ≥ 7. The smallest integer satisfying this is 7.</p>

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<p class="prompt">3. The temperature must stay between 65°F and 75°F, inclusive, for a chemical reaction. Write this as a compound inequality, where T is the temperature.</p>

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<p><span class="label">Explanation:</span> "Between 65 and 75, inclusive" means both endpoints are included: 65 ≤ T ≤ 75.</p>

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<p class="prompt">4. A student's test average must be within 5 points of 85 to pass. Write this as an absolute value inequality, where x is the test average.</p>

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<p><span class="label">Explanation:</span> "Within 5 points of 85" means the distance between x and 85 is at most 5: |x − 85| ≤ 5.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Forgetting to flip the inequality symbol when multiplying or dividing by a negative number.</strong> This is the single most common inequality error, always double-check the sign of what you're multiplying or dividing by.</li>

<li><strong>Applying an operation to only part of a compound inequality.</strong> Whatever you do to one part of a < expression < b, you must do to all three parts, the left side, the middle, and the right side.</li>

<li><strong>Treating |expression| > k the same way as |expression| < k.</strong> A "less than" absolute value inequality produces one connected range (an "and" situation). A "greater than" absolute value inequality produces two separate, non-overlapping rays (an "or" situation). Mixing these up is a very common error.</li>

<li><strong>Misreading inclusive versus exclusive language.</strong> "At least" and "at most" include the boundary value (≥ or ≤); "more than" and "less than" alone do not (> or <).</li>

<li><strong>Using an open circle when the boundary value is included, or vice versa, on a number line graph.</strong> A closed (filled) circle means the endpoint is included (≤ or ≥); an open circle means it isn't (< or >).</li>

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<h2 id="faq">Frequently Asked Questions</h2>

<h3>How can I remember when to flip the inequality sign?</h3>

<p>The rule applies only when multiplying or dividing by a negative number, never when adding or subtracting. A quick check: test a simple case like −2 < 3; if you divide both sides by −1 without flipping, you'd get 2 < −3, which is false, confirming the symbol must flip.</p>

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<h3>What's the difference between a compound inequality and an absolute value inequality?</h3>

<p>They're closely related: |expression| < k is actually a shortcut way of writing the compound inequality −k < expression < k. The absolute value notation is just a more compact way to express the same "within a certain distance" idea.</p>

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<h3>Why do "greater than" absolute value inequalities split into "or" statements instead of "and"?</h3>

<p>|expression| > k means the expression is more than k units away from 0 in either direction, positive or negative, so it could be greater than k OR less than −k. These two possibilities don't overlap, which is why they're joined with "or" rather than existing as one continuous range.</p>

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<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-inequalities-quiz-1">Inequalities quizzes</a> in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-inequalities-quiz-1">Practice Inequalities Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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