Algebra 1 Linear Inequalities in One Variable Worksheet Solutions (PDF)

Algebra 1 math worksheet solutions: Linear Inequalities in One Variable – Maths PDF

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This linear inequalities answers for Algebra 1 covers linear inequalities in one variable (Common Core A-REI.B.3, A-CED.A.1), with a free PDF to print.

Use this key to check your work on the linear inequalities worksheet. Every solution shows the steps, and the final answer appears in bold. If your answer differs, find the first line where your work and the key disagree, then redo the problem from there without looking.

Algebra 1 · Common Core State Standards: A-REI.B.3, A-CED.A.1

Linear inequalities answers: exercises 1 to 10

Exercise 1 – Graph x greater than 3

Open circle at 3.

Because 3 is not included, use an open circle at 3 and shade to the right.

Open circle at 3, ray to the right

Exercise 2 – Graph with a closed circle

Closed circle at -2.

Because -2 is included, use a closed circle at -2 and shade to the left.

Closed circle at -2, ray to the left

Exercise 3 – Write from a graph

The circle at 1 is closed, so 1 is included. The ray points right, to greater values.

$x \ge 1$

Exercise 4 – Add and compare

Subtract 5 from both sides: $x < 7$.

$x < 7$

Exercise 5 – Subtract on both sides

Add 4 to both sides: $x \ge 3$.

$x \ge 3$

Exercise 6 – Divide by a positive

Divide both sides by 3. The symbol stays the same: $x > 6$.

$x > 6$

Exercise 7 – Divide by a negative

Divide both sides by $-2$ and reverse the symbol: $x \ge -5$.

$x \ge -5$

Exercise 8 – Divide by a negative fraction form

Multiply both sides by $-3$ and reverse the symbol: $x > -12$.

$x > -12$

Exercise 9 – Two-step

Add 5: $2x \ge 14$. Divide by 2: $x \ge 7$.

$x \ge 7$

Exercise 10 – Two-step with a negative

Subtract 4: $-3x > 15$.

Divide by $-3$ and reverse the symbol: $x < -5$.

$x < -5$

Exercises 11 and up

Exercise 11 – Variables on both sides

Subtract $3x$: $2x + 2 \le 12$. Subtract 2: $2x \le 10$.

Divide by 2: $x \le 5$.

$x \le 5$

Exercise 12 – Distribute

Distribute: $4x – 8 < 2x + 6$. Subtract $2x$: $2x - 8 < 6$.

Add 8: $2x < 14$. Divide by 2: $x < 7$.

$x < 7$

Exercise 13 – Fraction coefficient

Add 1: $\frac{2}{3}x \ge 6$.

Multiply by $\frac{3}{2}$ (positive, so the symbol stays): $x \ge 9$.

$x \ge 9$

Exercise 14 – And compound

Closed circle at -1, open circle at 3.

Add 2 to all three parts: $-3 \le 3x < 9$.

Divide all parts by 3: $-1 \le x < 3$.

$-1 \le x < 3$

Exercise 15 – Or compound

Two separate rays.

First inequality: $x < -2$. Second inequality: $x \ge 4$.

$x < -2 \text{ or } x \ge 4$

Exercise 16 – Smoothies

Total cost cannot exceed $60: $4.50s \le 60$.

Divide by 4.50: $s \le 13.\overline{3}$.

She can buy only a whole number of smoothies, and the table shows 13 cost $58.50 while 14 cost $63.

13 smoothies

Exercise 17 – Saving for a bike

At least $200: $44 + 12n \ge 200$.

Subtract 44: $12n \ge 156$. Divide by 12: $n \ge 13$.

13 lawns

Exercise 18 – Elevator limit

Total weight: $180 + 45b \le 1000$.

Subtract 180: $45b \le 820$. Divide by 45: $b \le 18.\overline{2}$.

Check: 18 boxes weigh $180 + 810 = 990$ pounds, but 19 boxes weigh 1,035 pounds.

18 boxes

Exercise 19 – Phone plans

$30 + 0.10t < 45$. Subtract 30: $0.10t < 15$. Divide by 0.10: $t < 150$.

Fewer than 150 texts

Exercise 20 – Average score

Scores of 90 or more work.

$rac{82 + 90 + 78 + x}{4} \ge 85$. Multiply by 4: $250 + x \ge 340$.

Subtract 250: $x \ge 90$.

At least 90

Exercise 21 – Cooler temperature

Safe for 0 to 4 hours.

$34 \le 34 + 1.5h \le 40$. Subtract 34 from every part: $0 \le 1.5h \le 6$.

Divide by 1.5: $0 \le h \le 4$.

0 to 4 hours

Key facts behind this linear inequalities answers

Key facts

Use an open circle for < and >, and a closed circle for ≤ and ≥.
Adding or subtracting the same number never changes the inequality symbol.
Multiplying or dividing both sides by a negative number reverses the symbol.
An and inequality is true between endpoints; an or inequality is true on two rays.
“At least” means ≥ and “at most” means ≤.

How to use this linear inequalities answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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