7th Grade Two-Step Inequalities Worksheet Solutions (PDF)

7th Grade math worksheet solutions: Two-Step Inequalities – Maths PDF

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This grade 7 inequalities answers for 7th Grade covers two-step inequalities (Common Core 7.EE.B.4b), with a free PDF to print.

Use this answer key to check your work on the inequalities worksheet. Every solution shows the steps, and the final answer appears in bold. If your answer is different, compare each step and look for the first place where the work changes.

Grade 7 · Common Core State Standards: 7.EE.B.4b

Grade 7 inequalities answers: exercises 1 to 10

Exercise 1 – Greater than

Subtract 4 from both sides: \(5x > 25\).

Divide both sides by 5: \(x > 5\).

Check a value in the solution set: \(x = 6\) gives \(34 > 29\), which is true.

\(x > 5\)

Graph of x > 5 with an open circle.

Exercise 2 – At most

Add \(5\) to both sides: \(3x\le15\)

Divide both sides by \(3\): \(x\le5\)

Check a value in the solution set: \(x=4\) gives \(7\le10\), which is true.

\(x\le5\)

Graph of x ≤ 5: closed circle.

Exercise 3 – Divide by 2

Subtract \(1\) from both sides: \(\frac{1}{2}x\ge3\)

Divide both sides by \(\frac{1}{2}\): \(x\ge6\)

Check a value in the solution set: \(x=7\) gives \(4\frac{1}{2}\ge4\), which is true.

\(x\ge6\)

Graph of x ≥ 6: closed circle.

Exercise 4 – Reverse the symbol

Subtract \(4\) from both sides: \(-2x< 6\)

Divide both sides by \(-2\) and reverse the inequality symbol: \(x>-3\)

Check a value in the solution set: \(x=-2\) gives \(8< 10\), which is true.

\(x>-3\)

Graph of x > −3: open circle.

Exercise 5 – With parentheses

Divide both sides by 4: \(x-1>3\).

Add 1 to both sides: \(x>4\).

Check: \(x=5\) gives \(4(4)=16>12\), which is true.

\(x>4\)

Graph of x > 4 with an open circle.

Exercise 6 – Decimals

Add \(2\) to both sides: \(\frac{1}{2}x\le3\frac{1}{2}\)

Divide both sides by \(\frac{1}{2}\): \(x\le7\)

Check a value in the solution set: \(x=6\) gives \(1\le1\frac{1}{2}\), which is true.

\(x\le7\)

Graph of x ≤ 7: closed circle.

Exercise 7 – Negative at most

Add \(6\) to both sides: \(-3x\ge15\)

Divide both sides by \(-3\) and reverse the inequality symbol: \(x\le-5\)

Check a value in the solution set: \(x=-6\) gives \(12\ge9\), which is true.

\(x\le-5\)

Graph of x ≤ −5: closed circle.

Exercise 8 – Fraction coefficient

Subtract \(2\) from both sides: \(\frac{1}{3}x< 3\)

Divide both sides by \(\frac{1}{3}\): \(x< 9\)

Check a value in the solution set: \(x=8\) gives \(4\frac{2}{3}< 5\), which is true.

\(x< 9\)

Graph of x < 9: open circle.

Exercise 9 – Subtract from a constant

Subtract \(5\) from both sides: \(-x>-3\)

Divide both sides by \(-1\) and reverse the inequality symbol: \(x< 3\)

Check a value in the solution set: \(x=2\) gives \(3>2\), which is true.

\(x< 3\)

Graph of x < 3: open circle.

Exercise 10 – Negative outside

Divide both sides by \(-4\) and reverse the symbol: \(x+2\ge -3\).

Subtract 2: \(x\ge -5\).

Check: \(x=0\) gives \(-8\le 12\), which is true.

\(x\ge -5\)

Graph of x ≥ -5 with a closed circle.

Exercises 11 and up

Exercise 11 – Is it a solution?

\(3(7)-4=17>15\) is true.

\(3(6)-4=14>15\) is false.

7 is a solution; 6 is not

Exercise 12 – Read the graph

The circle is closed at \(-2\) and the shading goes left.

The inequality is \(x\le -2\).

Solutions: \(-2\) and \(-5\). Not a solution: \(0\).

\(x\le -2\)

Exercise 13 – Ticket budget

\(6t+4\le 40\).

\(6t\le 36\), so \(t\le 6\).

\(t\le 6\); at most 6 tickets

Exercise 14 – Earning goal

\(12h+13\ge 85\).

\(12h\ge 72\), so \(h\ge 6\).

\(h\ge 6\); at least 6 hours

Exercise 15 – Elevator limit

\(45b+150\le 1000\).

\(45b\le 850\), so \(b\le 18.9\) (about).

At most 18 boxes

Exercise 16 – Cooling down

\(50-4h< 32\).

\(-4h<-18\).

Divide by \(-4\) and reverse the symbol: \(h>4.5\).

\(h>4.5\); after 4.5 hours, so by hour 5 it is below 32°F

Temperature by hour.

Exercise 17 – Savings plan

\(120-15w\ge 30\).

\(-15w\ge -90\).

Divide by \(-15\) and reverse: \(w\le 6\)

\(w\le 6\); at most 6 weeks

Money left.

Exercise 18 – Taxi fare

\(2.25m+3.5\le 20\).

\(2.25m\le 16.5\), so \(m\le 7.33\) (about).

At most 7 miles

Exercise 19 – Phone minutes

\(0.08m+15< 25\).

\(0.08m< 10\), so \(m< 125\).

\(m< 125\); fewer than 125 minutes

m < 125 on a number line.

Exercise 20 – Car wash goal

\(3.5c+45>150\).

\(3.5c>105\), so \(c>30\).

Because c must be a whole number greater than 30, the least is 31.

\(c>30\); at least 31 cars

Key facts behind this grade 7 inequalities answers

Key facts

Solve \(px + q > r\) with the same steps as an equation.
Reverse the symbol when you multiply or divide both sides by a negative number.
Use an open circle for \(<\) and \(>\), and a closed circle for \(\le\) and \(\ge\).
Shade the side of the number line that contains the solutions.
Interpret the solution set in the context of the problem.

How to use this grade 7 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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