This algebra 1 linear worksheet answer key solves all 22 problems of the linear and compound inequalities worksheet step by step.
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 1 Linear Worksheet Answer Key: problems 1–5
Problem 1: One-step graph
Subtract 4: \(3x>15\). Divide by 3: \(x>5\). Interval notation: (5, ∞).
Problem 2: Closed circle
Add 7: \(5x\le35\). Divide by 5: \(x\le7\). Interval notation: (-∞, 7].
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.
Algebra 1 Linear Worksheet Answer Key: problems 6–10
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: \(x< 10\).
Algebra 1 Linear Worksheet Answer Key: problems 11–15
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).
Problem 13: Compound with and
Subtract 1 from all three parts: \(-4< 2x\le8\). Divide by 2: \(-2< x\le4\). Interval notation: (-2, 4].
Problem 14: Compound with or
First part: \(2x<-4\), so \(x<-2\). Second part: \(3x\ge9\), so \(x\ge3\).
Solution: \(x<-2\text{ or }x\ge3\).
Problem 15: Three-part inequality
Add 2 to all parts: \(6\le3x< 15\). Divide by 3: \(2\le x< 5\). Interval notation: [2, 5).
Algebra 1 Linear Worksheet Answer Key: problems 16–20
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 \(\$45\).
Algebra 1 Linear Worksheet Answer Key: problems 21–22
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: \(95\le s\le100\).
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 \(35+5(23)=150\).
How to use this algebra 1 linear worksheet answer key
Read this algebra 1 linear worksheet answer key after trying each problem yourself, and redo any problem you missed.
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