Unit 5 Equations and Inequalities Warm-Up- Practice Problems

What This Post Actually Is

A collection of practice problems for Unit 5 equations and inequalities. No frills, no colorful motivational speeches. Just problems, answers, and enough explanation to keep you from getting stuck for hours.

If you're a teacher looking for a warm-up packet, copy these. If you're a student trying to survive homework, work through them in order. Either way, the answers are at the bottom.

Why You Can't Skip This Unit

Equations and inequalities show up in every math class after this. Algebra 2, precalc, calculus—they all assume you can solve for x without having a meltdown. If you half-ass this unit, you're just delaying the inevitable.

Do the work now or do it later when there's more pressure. Your call.

Quick Reference: Equation Types in Unit 5

Before you start, know what you're dealing with:

• One-step equations — Addition, subtraction, multiplication, or division. One operation gets you the answer.
• Two-step equations — Two operations. Usually involves a coefficient first, then a constant.
• Multi-step equations — Variables on both sides. Requires distributing and combining like terms.
• Inequalities — Same as equations but with <, >, ≤, or ≥ instead of =. One extra rule: flip the sign when multiplying or dividing by negative.

One-Step Equation Practice

These are the easy ones. If these trip you up, go back and relearn how to add, subtract, multiply, or divide integers.

Problems 1-5

Solve for x:

1. x + 7 = 15
2. x - 4 = 11
3. 3x = 24
4. x/5 = 3
5. x + (-9) = -2

Solutions

1. x = 8 — Subtract 7 from both sides
2. x = 15 — Add 4 to both sides
3. x = 8 — Divide both sides by 3
4. x = 15 — Multiply both sides by 5
5. x = 7 — Add 9 to both sides (or subtract -9)

Two-Step Equation Practice

Two steps. Undo the coefficient first, then the constant. That's the order.

Problems 6-10

Solve for x:

1. 2x + 5 = 17
2. 4x - 3 = 21
3. 3x + 8 = 2
4. 5x - 12 = 33
5. 7 + 6x = 43

Solutions

1. x = 6 — Subtract 5 (x = 12), then divide by 2
2. x = 6 — Add 3 (4x = 24), then divide by 4
3. x = -2 — Subtract 8 (3x = -6), then divide by 3
4. x = 9 — Add 12 (5x = 45), then divide by 5
5. x = 6 — Subtract 7 (6x = 36), then divide by 6

Multi-Step Equation Practice

Now it gets real. Variables on both sides, distributive property, combining like terms. Don't skip steps or you'll get burned.

Problems 11-15

Solve for x:

1. 3x + 5 = 2x + 12
2. 5x - 7 = 3x + 9
3. 4(x + 2) = 3x + 11
4. 2x + 3 = 7x - 22
5. 6(x - 1) = 4x + 10

Solutions

1. x = 7 — Subtract 2x from both sides (x + 5 = 12), subtract 5
2. x = 8 — Subtract 3x (2x - 7 = 9), add 7 (2x = 16), divide by 2
3. x = 3 — Distribute: 4x + 8 = 3x + 11, subtract 3x (x + 8 = 11), subtract 8
4. x = 5 — Subtract 2x from both sides (3 = 5x - 22), add 22 (25 = 5x), divide by 5
5. x = 8 — Distribute: 6x - 6 = 4x + 10, subtract 4x (2x - 6 = 10), add 6 (2x = 16), divide by 2

Inequality Practice

Same process as equations. The only difference: if you multiply or divide by a negative number, flip the sign. That's it. That's the whole rule.

Problems 16-20

Solve each inequality and graph on a number line:

1. x + 4 > 9
2. 3x ≤ 18
3. 2x - 5 > 7
4. -4x < 12
5. x/3 + 2 ≥ 5

Solutions

1. x > 5 — Subtract 4. Open circle at 5, shade right.
2. x ≤ 6 — Divide by 3. Closed circle at 6, shade left.
3. x > 6 — Add 5 (2x > 12), divide by 2. Open circle at 6, shade right.
4. x > -3 — Divide by -4, FLIP THE SIGN. Open circle at -3, shade right.
5. x ≥ 9 — Subtract 2 (x/3 ≥ 3), multiply by 3. Closed circle at 9, shade right.

Common Mistakes to Avoid

These are the errors I see every year. Don't be that student.

• Forgetting to distribute — 3(x + 2) = 3x + 6, not 3x + 2. Always multiply every term inside the parentheses.
• Dropping negative signs — -x = 5 means x = -5. The negative doesn't just disappear.
• Forgetting to flip the inequality — This will cost you points on every test if you don't drill it.
• Combining unlike terms — x + x = 2x. That's fine. x + x² doesn't simplify. Know the difference.
• Rushing through checking your work — Plug your answer back in. Takes 10 seconds and catches most mistakes.

How to Check Your Answers (The Right Way)

Substitute your solution back into the original equation. If you got x = 5, plug in 5 where x was. Does it work? You're done. Doesn't work? Go back and find where you messed up.

Example: Check x = 6 in 2x + 5 = 17

2(6) + 5 = 12 + 5 = 17 ✓

That's it. That's the whole process. No excuse for not checking.

Word Problem Practice

Translate the words into math. Here's how:

• "More than" or "increased by" → +
• "Less than" or "decreased by" → -
• "Times" or "product of" → ×
• "Divided by" or "quotient" → ÷
• "Is" or "equals" → =

Problems 21-23

1. Three times a number minus 7 equals 20. What is the number?
2. The perimeter of a rectangle is 30 cm. The length is 3 more than twice the width. Find the width.
3. Sarah has $45. She buys a book for x dollars and still has $32. How much was the book?

Solutions

1. x = 9 — 3x - 7 = 20. Add 7 (3x = 27), divide by 3.
2. Width = 4 cm — Let width = w. Length = 2w + 3. Perimeter: 2(w + 2w + 3) = 30. Simplify: 2(3w + 3) = 30, 6w + 6 = 30, 6w = 24, w = 4.
3. $13 — 45 - x = 32. Subtract 32 from 45: x = 13.

Quick Comparison: Equations vs. Inequalities

Getting Started Checklist

Before you move on to the next unit, make sure you can do all of this without checking notes:

• Solve any one-step equation
• Solve any two-step equation
• Solve equations with variables on both sides
• Distribute and solve multi-step equations
• Solve inequalities including negative coefficients
• Graph inequalities on a number line
• Check your solutions by substitution
• Translate word problems into equations

Can't do all eight? That's your homework. Go back and practice the ones you missed.

Answers at a Glance

Problems 1-5: 8, 15, 8, 15, 7

Problems 6-10: 6, 6, -2, 9, 6

Problems 11-15: 7, 8, 3, 5, 8

Problems 16-20: x > 5, x ≤ 6, x > 6, x > -3, x ≥ 9

Problems 21-23: 9, 4 cm, $13

That's the unit. Now go do your actual homework.