How to Solve: ACT Math – Linear Equations and Inequalities

How to Solve: ACT Math – Linear Equations and Inequalities

Introduction

Mastering linear equations and inequalities unlocks 10-15% of your ACT Math score—that’s 6-9 questions you will see on test day. Whether you’re calculating a phone plan’s cost, figuring out when two trains meet, or solving for x in a word problem, these skills turn confusing questions into easy points.

What You Need To Know First

Before diving in, make sure you’re solid on: 1. Order of operations (PEMDAS/BODMAS) – Parentheses, exponents, multiply/divide, add/subtract. 2. Solving for one variable – Isolating x in simple equations like 3x + 5 = 20. 3. Number line basics – Understanding positive/negative numbers and inequalities (>, <, ≥, ≤).

If any of these feel shaky, pause and review them first—this guide builds on them.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Solving Linear Equations (One Variable)

Goal: Find the value of x that makes the equation true.

1. Simplify both sides – Combine like terms (e.g., 3x + 2x = 5x).
2. Move variables to one side – Use addition/subtraction to get all x terms on one side.
3. Move constants to the other side – Use addition/subtraction to isolate the x term.
4. Divide by the coefficient – Solve for x by dividing both sides by the number in front of x.
5. Check your answer – Plug x back into the original equation to verify.

Solving Linear Inequalities

Goal: Find all values of x that make the inequality true.

1. Solve like an equation – Follow steps 1-4 from above.
2. Flip the inequality sign if you multiply/divide by a negative number – This is the only difference from equations!
3. Graph the solution (if needed) – Draw a number line with an open/closed circle and shade the correct side.
4. > or < → open circle (not included)
5. ≥ or ≤ → closed circle (included)
6. Check your answer – Pick a number in your solution set and plug it back in.

Solving Systems of Equations

Goal: Find the (x, y) point where two lines intersect.

Method 1: Substitution 1. Solve one equation for y (or x). 2. Substitute that expression into the other equation. 3. Solve for the remaining variable. 4. Plug back in to find the other variable.

Method 2: Elimination 1. Align both equations. 2. Add or subtract them to eliminate one variable. 3. Solve for the remaining variable. 4. Plug back in to find the other variable.

Worked Examples

Example 1 – Basic Linear Equation

Problem: Solve for x: 3x - 7 = 14

Step-by-Step: 1. Add 7 to both sides: 3x = 21 2. Divide both sides by 3: x = 7 3. Check: 3(7) - 7 = 21 - 7 = 14 ✓

What we did and why: We isolated x by undoing the operations around it. First, we got rid of the -7 by adding 7, then we divided by 3 to solve for x.

Example 2 – Medium Inequality

Problem: Solve for x: -2x + 5 ≤ 11

Step-by-Step: 1. Subtract 5 from both sides: -2x ≤ 6 2. Divide both sides by -2 (flip the inequality!): x ≥ -3 3. Graph: Closed circle at -3, shade to the right. 4. Check: Pick x = 0 → -2(0) + 5 = 5 ≤ 11 ✓

What we did and why: We treated it like an equation, but flipped the sign when dividing by a negative. The solution is all x values greater than or equal to -3.

Example 3 – Exam-Style System of Equations

Problem: If 2x + y = 8 and x - y = 1, what is the value of x?

Step-by-Step (Elimination Method): 1. Write both equations:
- 2x + y = 8
- x - y = 1 2. Add them to eliminate y:
- (2x + y) + (x - y) = 8 + 1
- 3x = 9 3. Solve for x: x = 3 4. Plug x = 3 into x - y = 1:
- 3 - y = 1 → y = 2 5. Check: 2(3) + 2 = 8 ✓ and 3 - 2 = 1 ✓

What we did and why: We used elimination to cancel out y and solve for x. Then we plugged x back in to find y. The solution is (3, 2).

Common Mistakes

Exam Traps

1-Minute Recap

(Spoken naturally, as if to a student the night before the exam.)

"Okay, listen up—this is your 1-minute crash course for ACT linear equations and inequalities.

1. Equations: Isolate x by undoing operations. Add/subtract first, then multiply/divide. Always check your answer!
2. Inequalities: Same as equations, but flip the sign if you multiply or divide by a negative. Graph it with open/closed circles.
3. Systems: Use substitution or elimination. For elimination, add or subtract to cancel a variable. For substitution, solve one equation for y and plug it into the other.
4. Slope-intercept form: y = mx + b. m is slope, b is y-intercept. Memorize this!
5. Watch for traps: Flip inequality signs, check for no solution/infinite solutions, and translate word problems carefully.

You’ve got this. Practice 3-5 problems tonight, and you’ll own this on test day. Now go crush it!