How to Solve Linear Equations and Inequalities | ACT Math Guide for Grades 9-12
Linear equations and inequalities form the foundation of algebra and are among the most frequently tested topics on the ACT Math section. Whether you’re solving for $$x$$ in a simple equation like $$2x + 5 = 13$$ or working through an inequality such as $$3x – 7 < 11$$, mastering these concepts is essential for ACT success. The good news? Once you understand the fundamental rules and strategies, these problems become straightforward and quick to solve—giving you more time for challenging questions.
ACT SCORE BOOSTER: Master This Topic for 2-4 Extra Points!
Linear equations and inequalities appear in 8-12 questions on every ACT Math section. Understanding these concepts thoroughly can add 2-4 points to your composite score. Let’s break it down with proven strategies that work!
🚀 Jump to ACT Strategy →📚 Understanding Linear Equations and Inequalities
A linear equation is an algebraic statement where two expressions are equal, containing variables raised only to the first power. For example, $$3x + 7 = 22$$ is a linear equation. Your goal is to isolate the variable to find its value.
A linear inequality is similar, but instead of an equals sign, it uses inequality symbols: $$<$$ (less than), $$>$$ (greater than), $$\leq$$ (less than or equal to), or $$\geq$$ (greater than or equal to). For example, $$2x – 5 > 9$$ is a linear inequality. The solution is typically a range of values rather than a single number.
Why this matters for the ACT: These problems test your ability to manipulate algebraic expressions systematically and logically. They appear in various contexts—from straightforward “solve for x” questions to word problems involving real-world scenarios. Mastering these concepts builds the foundation for more advanced algebra topics like systems of equations and quadratic functions.
Frequency on the ACT: You can expect 8-12 questions involving linear equations and inequalities on every ACT Math test. This represents approximately 13-20% of the entire math section, making it one of the highest-yield topics to master.
⚡ Quick Answer: The Essential Strategy
For Linear Equations: Use inverse operations to isolate the variable. Whatever you do to one side, do to the other. Always simplify first, then solve.
For Linear Inequalities: Follow the same rules as equations, BUT remember: when you multiply or divide by a negative number, flip the inequality sign!
📐 Key Rules & Properties
🔹 Properties of Equality (for Equations)
- Addition Property: If $$a = b$$, then $$a + c = b + c$$
- Subtraction Property: If $$a = b$$, then $$a – c = b – c$$
- Multiplication Property: If $$a = b$$, then $$a \cdot c = b \cdot c$$
- Division Property: If $$a = b$$ and $$c \neq 0$$, then $$\frac{a}{c} = \frac{b}{c}$$
🔹 Properties of Inequality
- Addition/Subtraction: You can add or subtract the same number from both sides without changing the inequality direction
- Multiplication/Division by Positive: Multiplying or dividing by a positive number keeps the inequality direction the same
- Multiplication/Division by Negative: ⚠️ CRITICAL: When multiplying or dividing by a negative number, flip the inequality sign!
🔹 Standard Solving Process
- Simplify both sides (distribute, combine like terms)
- Move variable terms to one side
- Move constant terms to the other side
- Isolate the variable by dividing or multiplying
- Check your answer (substitute back into original)
✅ Step-by-Step Examples
Example 1: Solving a Basic Linear Equation
Problem: Solve for $$x$$: $$4x – 9 = 23$$
Step 1: Identify what we have
We have the equation $$4x – 9 = 23$$ and need to find the value of $$x$$.
Step 2: Isolate the variable term
Add 9 to both sides to eliminate the constant on the left:
$$4x – 9 + 9 = 23 + 9$$
$$4x = 32$$
Step 3: Solve for x
Divide both sides by 4:
$$\frac{4x}{4} = \frac{32}{4}$$
$$x = 8$$
Step 4: Check the answer
Substitute $$x = 8$$ back into the original equation:
$$4(8) – 9 = 32 – 9 = 23$$ ✓
Answer: $$x = 8$$
⏱️ ACT Time: 30-45 seconds
Example 2: Variables on Both Sides
Problem: Solve for $$x$$: $$7x + 5 = 3x + 21$$
Step 1: Move all variable terms to one side
Subtract $$3x$$ from both sides:
$$7x – 3x + 5 = 3x – 3x + 21$$
$$4x + 5 = 21$$
Step 2: Move constant terms to the other side
Subtract 5 from both sides:
$$4x + 5 – 5 = 21 – 5$$
$$4x = 16$$
Step 3: Solve for x
Divide both sides by 4:
$$x = 4$$
Step 4: Verify
Left side: $$7(4) + 5 = 28 + 5 = 33$$
Right side: $$3(4) + 21 = 12 + 21 = 33$$ ✓
Answer: $$x = 4$$
⏱️ ACT Time: 45-60 seconds
Example 3: Solving a Linear Inequality
Problem: Solve for $$x$$: $$-3x + 8 > 20$$
Step 1: Isolate the variable term
Subtract 8 from both sides:
$$-3x + 8 – 8 > 20 – 8$$
$$-3x > 12$$
Step 2: Solve for x (CRITICAL STEP!)
Divide both sides by -3.
⚠️ Remember: When dividing by a negative, FLIP the inequality sign!
$$\frac{-3x}{-3} < \frac{12}{-3}$$ (Notice the $$>$$ became $$<$$)
$$x < -4$$
Step 3: Interpret the solution
The solution is all values of $$x$$ that are less than -4.
Examples: $$x = -5$$, $$x = -10$$, $$x = -4.1$$ all work.
$$x = -4$$ does NOT work (not less than -4).
Step 4: Check with a test value
Let’s try $$x = -5$$:
$$-3(-5) + 8 = 15 + 8 = 23$$, and $$23 > 20$$ ✓
Answer: $$x < -4$$
⏱️ ACT Time: 45-60 seconds
Example 4: Equation with Distribution
Problem: Solve for $$x$$: $$2(3x – 4) = 5x + 6$$
Step 1: Distribute
Apply the distributive property on the left side:
$$2 \cdot 3x – 2 \cdot 4 = 5x + 6$$
$$6x – 8 = 5x + 6$$
Step 2: Move variable terms to one side
Subtract $$5x$$ from both sides:
$$6x – 5x – 8 = 5x – 5x + 6$$
$$x – 8 = 6$$
Step 3: Isolate x
Add 8 to both sides:
$$x – 8 + 8 = 6 + 8$$
$$x = 14$$
Step 4: Verify
Left side: $$2(3(14) – 4) = 2(42 – 4) = 2(38) = 76$$
Right side: $$5(14) + 6 = 70 + 6 = 76$$ ✓
Answer: $$x = 14$$
⏱️ ACT Time: 60-75 seconds
⚠️ Common Mistakes to Avoid
❌ Mistake #1: Forgetting to Flip the Inequality Sign
Wrong: Solving $$-2x > 6$$ → $$x > -3$$
✓ Correct: $$-2x > 6$$ → $$x < -3$$ (flip when dividing by negative!)
❌ Mistake #2: Distributing Incorrectly
Wrong: $$3(x + 2) = 3x + 2$$
✓ Correct: $$3(x + 2) = 3x + 6$$ (multiply BOTH terms inside)
❌ Mistake #3: Not Combining Like Terms First
Wrong: Jumping straight to solving $$2x + 3x – 5 = 10$$ without simplifying
✓ Correct: First simplify to $$5x – 5 = 10$$, then solve
❌ Mistake #4: Sign Errors When Moving Terms
Wrong: $$x – 7 = 12$$ → $$x = 12 – 7 = 5$$
✓ Correct: $$x – 7 = 12$$ → $$x = 12 + 7 = 19$$ (add 7, don’t subtract!)
❌ Mistake #5: Dividing Only One Term
Wrong: $$2x + 6 = 14$$ → $$x + 6 = 7$$ (only divided $$2x$$ by 2)
✓ Correct: First subtract 6: $$2x = 8$$, then divide: $$x = 4$$
📝 ACT-Style Practice Questions
Test your understanding with these ACT-style problems. Try solving them on your own before checking the solutions!
Practice Question 1 (Basic)
If $$5x – 12 = 33$$, what is the value of $$x$$?
Show Solution
Solution:
$$5x – 12 = 33$$
Add 12 to both sides: $$5x = 45$$
Divide by 5: $$x = 9$$
✓ Answer: C) 9
Difficulty: Basic | Time: 30 seconds
Practice Question 2 (Intermediate)
For what value of $$x$$ is the equation $$3(2x – 5) = 4x + 7$$ true?
Show Solution
Solution:
$$3(2x – 5) = 4x + 7$$
Distribute: $$6x – 15 = 4x + 7$$
Subtract $$4x$$: $$2x – 15 = 7$$
Add 15: $$2x = 22$$
Divide by 2: $$x = 11$$
✓ Answer: E) 11
Difficulty: Intermediate | Time: 60 seconds
Practice Question 3 (Intermediate – Inequality)
Which of the following describes all solutions to the inequality $$-4x + 6 \leq 18$$?
Show Solution
Solution:
$$-4x + 6 \leq 18$$
Subtract 6: $$-4x \leq 12$$
Divide by -4 (flip the sign!): $$x \geq -3$$
⚠️ Key Point: When dividing by a negative number, the inequality sign flips from $$\leq$$ to $$\geq$$!
✓ Answer: B) $$x \geq -3$$
Difficulty: Intermediate | Time: 45 seconds
Practice Question 4 (Advanced)
If $$\frac{2x + 5}{3} = \frac{x – 1}{2}$$, what is the value of $$x$$?
Show Solution
Solution:
$$\frac{2x + 5}{3} = \frac{x – 1}{2}$$
Cross-multiply: $$2(2x + 5) = 3(x – 1)$$
Distribute: $$4x + 10 = 3x – 3$$
Subtract $$3x$$: $$x + 10 = -3$$
Subtract 10: $$x = -13$$
💡 ACT Tip: Cross-multiplication is the fastest method for equations with fractions on both sides!
✓ Answer: A) -13
Difficulty: Advanced | Time: 75 seconds
🎯 ACT Test-Taking Strategy for Linear Equations & Inequalities
⏱️ Time Management
- Basic equations: Aim for 30-45 seconds
- Multi-step equations: Allow 60-75 seconds
- Inequalities: Budget 45-60 seconds (extra time to check sign flips)
- Word problems: Allow 90-120 seconds for translation and solving
🎲 When to Skip and Return
Skip if you encounter:
- Equations with complex fractions that require multiple steps
- Word problems where you can’t immediately identify the equation
- Problems involving absolute values (these are trickier)
Mark it and come back after completing easier questions!
✅ Quick Checking Strategy
The 10-Second Check: Always substitute your answer back into the original equation. If both sides equal, you’re correct!
For inequalities: Pick a test value from your solution range and verify it satisfies the original inequality.
🎯 Guessing Strategy
If you must guess:
- Plug in the answer choices (start with B, C, or D—middle values)
- Eliminate obviously wrong answers (e.g., negative when the equation suggests positive)
- For inequalities, remember: dividing by negatives flips the sign (eliminates half the choices!)
⚠️ Common Trap Answers
Watch out for these ACT traps:
- The “forgot to flip” trap: For $$-2x > 6$$, they’ll offer $$x > -3$$ (wrong!) alongside $$x < -3$$ (correct)
- The “partial solution” trap: Solving $$2x = 8$$ but forgetting to divide, offering 8 as an answer
- The “sign error” trap: Offering the negative of the correct answer
- The “wrong operation” trap: Results from adding when you should subtract
💡 ACT Pro Tips & Tricks
🚀 Tip #1: Work Backwards with Answer Choices
When solving equations, you can often plug in the answer choices to see which one works. This is especially useful for complex equations or when you’re short on time. Start with choice C (the middle value) to eliminate answers efficiently.
⚡ Tip #2: The “Flip Sign” Memory Trick
Remember: “Negative operation, flip the relation.” Whenever you multiply or divide by a negative number in an inequality, flip the inequality sign. Write a big “FLIP!” on your scratch paper when you see a negative coefficient.
📊 Tip #3: Use the Number Line for Inequalities
When solving inequalities, quickly sketch a number line on your scratch paper. Mark your solution and test a value to verify. This visual check takes 5 seconds and prevents costly mistakes.
🎯 Tip #4: Simplify Before You Solve
Always combine like terms and distribute first. Trying to solve $$2x + 3x – 5 = 10$$ without simplifying to $$5x – 5 = 10$$ wastes time and increases error risk. Make simplification your automatic first step.
🧮 Tip #5: Calculator Smart Usage
Your calculator can verify answers quickly! After solving algebraically, use your calculator to check: plug in your answer and verify both sides equal. This 5-second check catches arithmetic errors.
📝 Tip #6: Show Your Work (Even on ACT)
Write out each step on your test booklet. This prevents skipping steps mentally (where errors occur) and lets you backtrack if you get stuck. Organized work = fewer mistakes = higher score.
❓ Frequently Asked Questions
Q1: What’s the difference between an equation and an inequality?
An equation uses an equals sign (=) and has one specific solution (or sometimes no solution or infinitely many). An inequality uses symbols like <, >, ≤, or ≥ and typically has a range of solutions. For example, $$x = 5$$ is an equation with one solution, while $$x > 5$$ is an inequality with infinitely many solutions (all numbers greater than 5).
Q2: Why do we flip the inequality sign when multiplying or dividing by a negative?
Think about it this way: 3 < 5 is true. If we multiply both sides by -1, we get -3 and -5. But -3 is actually greater than -5 (it’s closer to zero on the number line). So the relationship flips: -3 > -5. This happens every time you multiply or divide by a negative—the order reverses. This is one of the most tested concepts on the ACT, so memorize it!
Q3: Can I use my calculator to solve linear equations on the ACT?
Yes, but strategically! While you should solve algebraically (it’s faster), you can use your calculator to verify answers by plugging them back into the original equation. Some graphing calculators also have equation solvers, but learning to solve by hand is faster for simple linear equations. Save calculator time for more complex problems.
Q4: What if I get a result like 0 = 0 or 5 = 3 when solving?
Great question! If you get 0 = 0 (or any true statement like 3 = 3), the equation has infinitely many solutions—every value of x works. If you get a false statement like 5 = 3, the equation has no solution. On the ACT, answer choices might include “all real numbers” or “no solution” for these cases.
Q5: How do I handle fractions in linear equations?
You have two main strategies: (1) Clear the fractions by multiplying both sides by the least common denominator (LCD), or (2) Cross-multiply if you have one fraction on each side. For example, with $$\frac{x}{3} = \frac{2x-1}{5}$$, cross-multiply to get $$5x = 3(2x-1)$$. This eliminates fractions immediately and makes solving easier. Practice both methods to see which feels more natural!
✍️ Written by Irfan Mansuri
ACT Test Prep Specialist & Educator
IrfanEdu.com • United States
Irfan Mansuri is a dedicated ACT test preparation specialist with over 15 years of experience helping high school students achieve their target scores. As the founder of IrfanEdu.com, he has guided thousands of students through the ACT journey, with many achieving scores of 30+ and gaining admission to their dream colleges. His teaching methodology combines deep content knowledge with proven test-taking strategies, making complex concepts accessible and helping students build confidence. Irfan’s approach focuses not just on memorization, but on true understanding and strategic thinking that translates to higher scores.
🎓 Final Thoughts: Your Path to ACT Math Success
Mastering linear equations and inequalities is one of the highest-impact investments you can make in your ACT Math preparation. These concepts appear in 8-12 questions per test, and with the strategies you’ve learned today, you can solve them quickly and accurately—often in under 60 seconds each.
Remember the key principles: simplify first, use inverse operations systematically, and always flip the inequality sign when multiplying or dividing by a negative. Practice these problems daily, check your work by substituting answers back, and you’ll build the speed and confidence needed for test day.
Keep practicing, stay confident, and watch your ACT Math score improve! 🚀
Linear Equations and Inequalities Guide
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