Solving Quadratic and Absolute Value Inequalities | ACT Math Guide
Inequalities can feel intimidating at first, but once you understand the core techniques for solving quadratic and absolute value inequalities, they become manageable—and even predictable on the ACT. These problems test your ability to think critically about ranges of solutions rather than single values, a skill that appears frequently in the ACT prep resources and on test day. Whether you’re dealing with parabolas or absolute value graphs, mastering these inequality types will give you a significant advantage in the Intermediate Algebra section.
ACT SCORE BOOSTER: Master This Topic for 2-3 Extra Points!
Quadratic and absolute value inequalities appear in 5-8 questions on the ACT Math section. Understanding these thoroughly can add 2-3 points to your composite score. Let’s break it down with proven strategies that work!
🚀 Jump to ACT Strategy →📚 Understanding Inequalities in ACT Math
While equations ask you to find specific values where expressions are equal, inequalities require you to identify entire ranges of values that satisfy a condition. On the ACT, you’ll encounter two particularly important types: quadratic inequalities (involving $$x^2$$ terms) and absolute value inequalities (involving $$|x|$$ notation).
These problems test your understanding of number lines, interval notation, and graphical reasoning. According to the official ACT website, intermediate algebra questions make up approximately 15-20% of the Math section, and inequalities are a recurring theme within this category.
🔑 Key Concept
The fundamental difference between equations and inequalities is that inequalities describe solution sets (ranges) rather than discrete solutions. Your goal is to determine which values make the inequality true, then express that range using interval notation or a number line.
📐 Essential Methods for Solving Inequalities
📋 Quadratic Inequalities Method
- Rearrange to standard form: Get everything on one side so you have $$ax^2 + bx + c > 0$$ (or $$<$$, $$\geq$$, $$\leq$$)
- Find critical points: Solve the related equation $$ax^2 + bx + c = 0$$ using factoring, quadratic formula, or completing the square
- Test intervals: The critical points divide the number line into regions. Test a value from each region in the original inequality
- Write solution: Identify which intervals satisfy the inequality and express using interval notation
📋 Absolute Value Inequalities Method
For $$|x| < a$$ (where $$a > 0$$):
$$-a < x < a$$
For $$|x| > a$$ (where $$a > 0$$):
$$x < -a$$ or $$x > a$$
⚠️ Critical Rule: The same patterns apply to $$\leq$$ and $$\geq$$, but remember to use brackets [ ] instead of parentheses ( ) in interval notation to include the endpoints!
✅ Step-by-Step Solved Examples
Example 1: Quadratic Inequality
Solve: $$x^2 – 5x + 6 < 0$$
Step 1: Find the critical points
First, solve the related equation $$x^2 – 5x + 6 = 0$$. We can factor this:
$$(x – 2)(x – 3) = 0$$
So our critical points are $$x = 2$$ and $$x = 3$$.
Step 2: Identify the intervals
These critical points divide the number line into three regions:
- Region 1: $$x < 2$$
- Region 2: $$2 < x < 3$$
- Region 3: $$x > 3$$
Step 3: Test each interval
Let’s test a value from each region:
- Region 1 (test $$x = 0$$): $$0^2 – 5(0) + 6 = 6 > 0$$ ❌ (doesn’t satisfy $$< 0$$)
- Region 2 (test $$x = 2.5$$): $$(2.5)^2 – 5(2.5) + 6 = 6.25 – 12.5 + 6 = -0.25 < 0$$ ✅
- Region 3 (test $$x = 4$$): $$4^2 – 5(4) + 6 = 16 – 20 + 6 = 2 > 0$$ ❌
Step 4: Write the solution
✅ Solution: $$2 < x < 3$$ or in interval notation: $$(2, 3)$$
⏱️ ACT Time Estimate: 60-90 seconds if you can factor quickly
Example 2: Absolute Value Inequality (Less Than)
Solve: $$|2x – 5| < 7$$
Step 1: Apply the “less than” rule
For $$|A| < B$$, we write: $$-B < A < B$$
$$-7 < 2x - 5 < 7$$
Step 2: Solve the compound inequality
Add 5 to all three parts:
$$-7 + 5 < 2x - 5 + 5 < 7 + 5$$
$$-2 < 2x < 12$$
Divide all parts by 2:
$$-1 < x < 6$$
✅ Solution: $$-1 < x < 6$$ or in interval notation: $$(-1, 6)$$
⏱️ ACT Time Estimate: 30-45 seconds with practice
Example 3: Absolute Value Inequality (Greater Than)
Solve: $$|x + 3| \geq 4$$
Step 1: Apply the “greater than” rule
For $$|A| \geq B$$, we write two separate inequalities: $$A \leq -B$$ OR $$A \geq B$$
$$x + 3 \leq -4$$ OR $$x + 3 \geq 4$$
Step 2: Solve each inequality separately
First inequality: $$x + 3 \leq -4$$
Subtract 3: $$x \leq -7$$
Second inequality: $$x + 3 \geq 4$$
Subtract 3: $$x \geq 1$$
✅ Solution: $$x \leq -7$$ or $$x \geq 1$$
In interval notation: $$(-\infty, -7] \cup [1, \infty)$$
💡 Notice: We use brackets [ ] because the inequality includes “or equal to” ($$\geq$$). The union symbol $$\cup$$ means “or” in interval notation.
⏱️ ACT Time Estimate: 45-60 seconds
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🚀 Start ACT Practice Test Now →📝 Practice Questions
Test your understanding with these ACT-style practice problems. Click “Show Solution” to see detailed explanations.
Practice Question 1 Intermediate
Solve the inequality: $$x^2 + 2x – 8 > 0$$
Show Solution
✅ Correct Answer: A
Solution:
- Factor: $$(x + 4)(x – 2) = 0$$, so critical points are $$x = -4$$ and $$x = 2$$
- Test intervals:
- $$x = -5$$: $$(-5 + 4)(-5 – 2) = (-1)(-7) = 7 > 0$$ ✅
- $$x = 0$$: $$(0 + 4)(0 – 2) = (4)(-2) = -8 < 0$$ ❌
- $$x = 3$$: $$(3 + 4)(3 – 2) = (7)(1) = 7 > 0$$ ✅
- Solution: $$x < -4$$ or $$x > 2$$
Practice Question 2 Basic
Solve: $$|3x + 1| \leq 8$$
Show Solution
✅ Correct Answer: A
Solution:
- Apply the rule: $$-8 \leq 3x + 1 \leq 8$$
- Subtract 1 from all parts: $$-9 \leq 3x \leq 7$$
- Divide by 3: $$-3 \leq x \leq \frac{7}{3}$$
Practice Question 3 Advanced
For what values of $$x$$ is $$|2x – 3| > 5$$?
Show Solution
✅ Correct Answer: B
Solution:
- Apply the “greater than” rule: $$2x – 3 < -5$$ OR $$2x - 3 > 5$$
- First inequality: $$2x – 3 < -5$$ → $$2x < -2$$ → $$x < -1$$
- Second inequality: $$2x – 3 > 5$$ → $$2x > 8$$ → $$x > 4$$
- Solution: $$x < -1$$ or $$x > 4$$
💡 ACT Pro Tips & Tricks
🎯 Strategic Tips for ACT Success
✨ Remember the Sign Flip Rule
When multiplying or dividing an inequality by a negative number, you must flip the inequality sign. This is a common trap on the ACT! Always double-check your work when dealing with negative coefficients.
🎨 Visualize with Number Lines
When solving quadratic inequalities, quickly sketch a parabola or number line. Visual learners often find this faster than algebraic testing. Mark your critical points and shade the regions that satisfy the inequality.
⚡ Memorize the Absolute Value Patterns
$$|x| < a$$ means “between” (one interval)
$$|x| > a$$ means “outside” (two intervals)
This simple memory trick saves precious seconds on test day!
🔍 Watch for Boundary Points
Pay attention to whether the inequality uses $$<$$ or $$\leq$$. The difference determines whether you use parentheses ( ) or brackets [ ] in your answer. ACT answer choices often differ only in this detail!
🧮 Use Your Calculator Wisely
For quadratic inequalities, you can graph $$y = ax^2 + bx + c$$ on your calculator and visually identify where the graph is above or below the x-axis. This is especially helpful when factoring is difficult.
⏰ Test Smart, Not Hard
If you’re running short on time, you can test the answer choices by plugging in values. Pick a number from each interval in the answer choices and see which one satisfies the original inequality. This backup strategy can save you when algebra gets messy!
⚠️ Common Mistakes to Avoid
❌ Mistake #1: Forgetting to Flip the Inequality
When you multiply or divide by a negative number, the inequality sign must reverse. For example, if you have $$-2x > 6$$, dividing by $$-2$$ gives $$x < -3$$, NOT $$x > -3$$.
❌ Mistake #2: Confusing “And” vs “Or”
For $$|x| < a$$, the solution is $$-a < x < a$$ (one connected interval - "and").
For $$|x| > a$$, the solution is $$x < -a$$ OR $$x > a$$ (two separate intervals – “or”).
Mixing these up is one of the most common errors on the ACT!
❌ Mistake #3: Testing Only One Interval
For quadratic inequalities, you must test all intervals created by the critical points. Don’t assume the pattern—always verify each region!
🎯 ACT Test-Taking Strategy for Inequalities
⏱️ Time Allocation
Spend 60-90 seconds on basic absolute value inequalities and 90-120 seconds on quadratic inequalities. If a problem takes longer, mark it and return later—don’t let one question derail your timing.
🎲 When to Skip and Return
If you can’t factor the quadratic within 15 seconds, either use the quadratic formula quickly or skip and return. Don’t waste time struggling with difficult factoring when other questions might be easier.
🎯 Guessing Strategy
If you must guess on an absolute value inequality, remember: “less than” ($$<$$) typically gives you ONE interval (between two values), while "greater than" ($$>$$) gives you TWO intervals (outside the range). Eliminate answers that don’t match this pattern.
✅ Quick Check Method
After solving, plug in one value from your solution set into the original inequality. If it works, you’re likely correct. This 5-second check can catch sign errors and prevent careless mistakes.
🚨 Watch for These Trap Answers
- Answer choices with the inequality sign flipped
- Solutions using parentheses when brackets are needed (or vice versa)
- Switching “and” for “or” in absolute value problems
- Critical points themselves listed as solutions when they shouldn’t be included
🎥 Video Explanation: Solving Inequalities
Watch this detailed video explanation to master quadratic and absolute value inequalities with visual demonstrations and step-by-step guidance.
❓ Frequently Asked Questions
✍️ Written by Dr. Irfan Mansuri
Educational Content Creator & Competitive Exam Specialist
IrfanEdu.com • United States
Dr. Irfan Mansuri is a distinguished educational content creator with over 15 years of experience spanning high school, undergraduate, and postgraduate levels. As the founder of IrfanEdu.com, he has successfully guided thousands of students through competitive examinations, helping them achieve exceptional results and gain admission to their dream institutions.
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Start Your Free ACT Practice Test →Understanding Quadratic and Absolute Value Inequalities
Students often encounter challenges when working with inequalities that involve quadratic expressions or absolute values. This comprehensive guide breaks down these concepts into manageable steps, helping you develop confidence in solving these mathematical problems.
What Are Quadratic Inequalities?
A quadratic inequality presents itself when you compare a quadratic expression to zero or another value. You work with expressions like $$ax^2 + bx + c > 0$$ or similar variations using different inequality symbols. The goal involves finding all x-values that make the inequality true.
Step-by-Step Approach to Solving Quadratic Inequalities
Step 1: Rearrange the Inequality
Begin by moving all terms to one side of the inequality. You want to create a format where the quadratic expression sits on one side and zero appears on the other. For example, if you start with $$2x^2 \leq 3 – x$$, you rearrange it to $$2x^2 + x – 3 \leq 0$$.
Step 2: Identify the Boundary Points
You find the roots by solving the corresponding equation where the expression equals zero. These roots serve as critical boundary points. You can factor the quadratic expression when possible, or apply the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$.
Step 3: Create Test Intervals
The roots divide your number line into distinct intervals. You select a test value from each interval and substitute it into the original inequality. This process reveals whether the expression produces positive or negative values in that region.
Step 4: Determine the Solution Set
Based on your test results, you identify which intervals satisfy the inequality. Remember to include or exclude the boundary points depending on whether the inequality uses “or equal to” symbols.
Working with Absolute Value Inequalities
Absolute value represents the distance a number sits from zero on the number line. This distance always remains positive or zero, never negative. When you solve absolute value inequalities, you consider two scenarios based on the inequality type.
The “Less Than” Pattern
When you encounter $$|x| < a$$ where a represents a positive number, you translate this into a compound inequality: $$-a < x < a$$. This creates an "and" situation where x must fall between two values. For instance, $$|x - 3| \leq 5$$ becomes $$-5 \leq x - 3 \leq 5$$, which simplifies to $$-2 \leq x \leq 8$$.
The “Greater Than” Pattern
For inequalities like $$|x| > a$$ with positive a, you split this into two separate conditions: $$x > a$$ or $$x < -a$$. This creates an "or" situation. Consider $$|x + 2| > 4$$, which breaks into $$x + 2 > 4$$ or $$x + 2 < -4$$, giving you $$x > 2$$ or $$x < -6$$.
Essential Tips for Success
- Always isolate the absolute value expression before applying solution rules
- Watch for sign changes when you multiply or divide by negative numbers
- Use brackets [ ] when the inequality includes the boundary points
- Use parentheses ( ) when the inequality excludes the boundary points
- Check for impossible situations, such as absolute values less than negative numbers
Practical Example: Solving a Quadratic Inequality
Let’s solve $$x^2 – 4 > 0$$ step by step:
First, we find the roots by setting $$x^2 – 4 = 0$$. This factors as $$(x – 2)(x + 2) = 0$$, giving us $$x = 2$$ and $$x = -2$$.
These roots create three intervals: $$(-\infty, -2)$$, $$(-2, 2)$$, and $$(2, \infty)$$.
We test each interval:
- For $$x = -3$$: $$(-3)^2 – 4 = 5 > 0$$ ✓
- For $$x = 0$$: $$(0)^2 – 4 = -4 > 0$$ ✗
- For $$x = 3$$: $$(3)^2 – 4 = 5 > 0$$ ✓
The solution becomes $$(-\infty, -2) \cup (2, \infty)$$.
Recognizing Special Cases
You need to watch for situations where no solution exists. If you isolate an absolute value and find it must be less than a negative number, the inequality has no solution. Conversely, if an absolute value must be greater than a negative number, all real numbers satisfy the inequality.
Visualizing Solutions Graphically
Graphing provides powerful visual confirmation of your solutions. When you graph the functions on both sides of an inequality, the solution corresponds to where one graph sits above or below the other. Intersection points mark the boundary values of your solution intervals.
Real-World Applications
These inequality concepts appear frequently in practical situations. Engineers use them to determine acceptable tolerance ranges in manufacturing. Scientists apply them when analyzing measurement uncertainties. Business professionals employ them for profit optimization and cost analysis.
Building Your Problem-Solving Skills
Mastery comes through consistent practice. Start with simpler problems and gradually increase complexity. Always verify your solutions by substituting test values back into the original inequality. This habit builds confidence and catches potential errors early.
Understanding these inequality techniques opens doors to more advanced mathematical concepts. You develop analytical thinking skills that extend far beyond mathematics into logical reasoning and problem-solving in everyday life.
Quadratic and Absolute Value Inequalities ACT Math Guide act math practice test pdf
Quadratic and Absolute Value Inequalities ACT Math Guide act math practice test pdf
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