Linear and Compound Inequalities: Algebra 1 Math Worksheet Answer Key – PDF Download

Each solution shows every step, the graph or interval notation when requested, and a check. When a problem divides by a negative number, notice the line where the inequality sign reverses. If your answer differs, find the first line where your work and the key disagree.

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

Problem 1: One-step graph

Subtract 4: $3x>15$. Divide by 3: $x>5$. Interval notation: (5, ∞).

Graph of $x>5$.

Problem 2: Closed circle

Add 7: $5x\le35$. Divide by 5: $x\le7$. Interval notation: (-∞, 7].

Graph of $x\le7$.

Problem 3: Divide by a negative

Subtract 9: $-2x< 6$. Divide by $-2$ and reverse the sign: $x>-3$.

Check: $x=0$ gives $9< 15$, true.

Problem 4: Distribute and solve

Distribute: $4x-12\ge2x+6$. Subtract $2x$: $2x-12\ge6$. Add 12: $2x\ge18$. Divide by 2: $x\ge9$.

Problem 5: Variables on both sides

Add $3x$: $7>5x-8$. Add 8: $15>5x$. Divide by 5: $3>x$, which is $x< 3$.

Check: $x=0$ gives $7>-8$, true.

Problem 6: Fraction coefficient

Add 2: $\frac{x}{3}\le6$. Multiply by 3: $x\le18$.

Problem 7: Negative fraction

Multiply by $-\frac{3}{2}$ and reverse the sign: $x\le8\cdot(-\frac{3}{2})$, so $x\le-12$.

Check: $x=-12$ gives $8\ge8$, true.

Problem 8: Negative outside parentheses

Distribute: $-2x-8< 6$. Add 8: $-2x< 14$. Divide by $-2$ and reverse: $x>-7$.

Problem 9: Fractions on both sides

Multiply by 6: $3(x+1)\ge2(2x-1)$. Distribute: $3x+3\ge4x-2$. Subtract $3x$: $3\ge x-2$. Add 2: $5\ge x$, so $x\le5$.

Problem 10: Decimal inequality

Subtract $0.2x$: $0.4x-1.5< 2.5$. Add 1.5: $0.4x< 4$. Divide by 0.4: $x< 10$.

Problem 11: All or none

Distribute: $3x+6>3x+1$. Subtract $3x$: $6>1$, always true: all real numbers.
Subtract $2x$: $5<-1$, never true: no solution.

Problem 12: Read the graph

$x\ge-4$, interval [-4, ∞).
$x< 3$, interval (-∞, 3).

Graph of $x\ge-4$.

Problem 13: Compound with and

Subtract 1 from all three parts: $-4< 2x\le8$. Divide by 2: $-2< x\le4$. Interval notation: (-2, 4].

Graph of $-2< x\le 4$.

Problem 14: Compound with or

First part: $2x<-4$, so $x<-2$. Second part: $3x\ge9$, so $x\ge3$.

Solution: $x<-2\text{ or }x\ge3$.

Graph of $x<-2\text{ or }x\ge3$.

Problem 15: Three-part inequality

Add 2 to all parts: $6\le3x< 15$. Divide by 3: $2\le x< 5$. Interval notation: [2, 5).

Graph of $2\le x< 5$.

Problem 16: Overlap of two conditions

First: $x>-2$. Second: $x\le5$. Both must hold, so the solution is $-2< x\le5$.

Problem 17: School store budget

$18.50+6.25t\le50$. Subtract 18.50: $6.25t\le31.50$. Divide: $t\le5.04$.

Diego can buy at most 5 T-shirts. Check: $18.50+6.25(5)=49.75$, but 6 shirts cost $\$56.00$.

Problem 18: Weekend job

$12h-15\ge150$. Add 15: $12h\ge165$. Divide: $h\ge13.75$.

She must work at least 13.75 hours (13 hours 45 minutes). Check: $12(13.75)-15=150$.

Problem 19: Elevator limit

$180+35b\le1200$. Subtract 180: $35b\le1020$. Divide: $b\le29.14$.

At most 29 boxes. Check: $180+35(29)=1195$, while 30 boxes give $1230$.

Problem 20: Car rental

$30+0.20m< 45$. Subtract 30: $0.20m< 15$. Divide: $m< 75$.

A is cheaper for fewer than 75 miles. At 75 miles both cost $\$45$.

Problem 21: Test average

$\frac{88+92+85+s}{4}\ge90$. Multiply by 4: $265+s\ge360$. Subtract: $s\ge95$.

With the maximum of 100: $95\le s\le100$.

Graph of $95\le x\le 100$.

Problem 22: Saving up

$100\le35+5w\le150$. Subtract 35: $65\le5w\le115$. Divide by 5: $13\le w\le23$.

From week 13 to week 23. Check: $35+5(13)=100$ and $35+5(23)=150$.

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