Key takeaway: This grade 7 two worksheet answer key gives the solution set, the circle type, and a test value for each of the 20 problems. Word problems end with a sentence that reads the answer in context. Wherever a problem flips the sign, the key marks that step, so you can compare timing.
Compare the direction of your sign first, since a missed flip reverses the whole graph. Then check whether your circle should be open or closed.
Each of the twenty solutions below shows every step, a quick check with a number from the solution set, and the graph where one is needed. Compare your work step by step and watch for the sign flip when dividing by a negative number.
Grade 7 Two Worksheet Answer Key: problems 1–5
1. Solve and graph
Subtract 3 from both sides: \(2x>8\).
Divide both sides by 2: \(x>4\).
Check \(x=5\): \(2(5)+3=13>11\), true.
Answer: \(x>4\) (open circle at 4, shade right).
2. Greater than
Subtract 2 from both sides: \(3x>15\).
Divide both sides by 3: \(x>5\).
Check \(x=6\): \(3(6)+2=20>17\), true.
Answer: \(x>5\) (open circle at 5, shade right).
3. Subtract, then divide
Add 5 to both sides: \(3x< 15\).
Divide both sides by 3: \(x< 5\).
Check \(x=0\): \(3(0)-5=-5< 10\), true.
Answer: \(x< 5\) (open circle at 5, shade left).
4. Less than
Add 6 to both sides: \(4x< 16\).
Divide both sides by 4: \(x< 4\).
Check \(x=3\): \(4(3)-6=6< 10\), true. Check \(x=4\): \(4(4)-6=10< 10\) is false, so 4 is not included.
Answer: \(x< 4\) (open circle at 4, shade left).
5. Include the endpoint
Subtract 7: \(4x\ge 16\).
Divide by 4: \(x\ge 4\).
Check \(x=5\): \(4(5)+7=27\ge 23\), true. Check \(x=3\): \(19\ge 23\) is false, so 3 is not in the set.
Answer: \(x\ge 4\)
Grade 7 Two Worksheet Answer Key: problems 6–10
6. Include the endpoint, again
Subtract 9 from both sides: \(2x\ge -6\).
Divide both sides by 2: \(x\ge -3\).
Check \(x=0\): \(2(0)+9=9\ge 3\), true. Check \(x=-4\): \(2(-4)+9=1\ge 3\) is false.
Answer: \(x\ge -3\) (closed circle at \(-3\), shade right).
7. At most
Add 2: \(5x\le 20\).
Divide by 5: \(x\le 4\).
Answer: \(x\le 4\) (closed circle at 4, shade left).
8. Test the values
Add 4 to both sides: \(3x\le 15\).
Divide both sides by 3: \(x\le 5\).
Test \(x=3\): \(3(3)-4=5\le 11\), true. Test \(x=5\): \(3(5)-4=11\le 11\), true. Test \(x=6\): \(3(6)-4=14\le 11\), false.
Answer: \(x\le 5\); the values 3 and 5 are solutions, 6 is not.
9. A fraction coefficient
Subtract 3: \(\frac{x}{2}>4\).
Multiply both sides by 2: \(x>8\).
Check \(x=10\): \(\frac{10}{2}+3=8>7\), true.
Answer: \(x>8\)
10. Divide by five
Add 1 to both sides: \(\frac{x}{5}\le 3\).
Multiply both sides by 5: \(x\le 15\).
Check \(x=10\): \(\frac{10}{5}-1=1\le 2\), true.
Answer: \(x\le 15\)
Grade 7 Two Worksheet Answer Key: problems 11–15
11. Dividing by a negative
Subtract 1: \(-2x< 8\).
Divide by \(-2\) and reverse the sign: \(x>-4\).
Check \(x=0\): \(-2(0)+1=1< 9\), true.
Answer: \(x>-4\) (open circle at \(-4\), shade right).
12. Flip the sign
Subtract 3 from both sides: \(-4x>16\).
Divide both sides by \(-4\) and reverse the sign: \(x< -4\).
Check \(x=-5\): \(-4(-5)+3=23>19\), true.
Answer: \(x< -4\) (open circle at \(-4\), shade left).
13. Negative coefficient, closed circle
Add 4: \(-3x\ge 9\).
Divide by \(-3\) and reverse the sign: \(x\le -3\).
Check \(x=-4\): \(-3(-4)-4=8\ge 5\), true.
Answer: \(x\le -3\): a closed circle at \(-3\) with the ray shaded to the left.
14. Negative fraction
Add 3 to both sides: \(-\frac{1}{2}x\le 5\).
Multiply both sides by \(-2\) and reverse the sign: \(x\ge -10\).
Check \(x=0\): \(-\frac{1}{2}(0)-3=-3\le 2\), true. Check \(x=-12\): \(-\frac{1}{2}(-12)-3=3\le 2\) is false.
Answer: \(x\ge -10\)
15. School store budget
Inequality: \(12+9s\le 50\).
Subtract 12: \(9s\le 38\). Divide by 9: \(s\le 4\frac{2}{9}\).
Jamal cannot buy part of a shirt. Four shirts cost \(12+36=\$48\); five would cost \(\$57\).
Answer: at most 4 shirts.
Grade 7 Two Worksheet Answer Key: problems 16–20
16. Dog walking goal
Inequality: \(12h+35\ge 200\).
Subtract 35: \(12h\ge 165\). Divide by 12: \(h\ge 13.75\).
She must walk a whole number of hours. Thirteen hours give \(12(13)+35=\$191\), which is too little; fourteen give \(12(14)+35=\$203\).
Answer: at least 14 hours.
17. Delivery van
Inequality: \(220+40b\le 1000\).
Subtract 220: \(40b\le 780\). Divide by 40: \(b\le 19.5\).
Boxes come whole. Nineteen boxes weigh \(220+40(19)=980\) pounds; twenty would weigh \(1,020\) pounds, which is too much.
Answer: at most 19 boxes.
18. Freezer temperature
Inequality: \(10-3h<-8\).
Subtract 10: \(-3h<-18\).
Divide by \(-3\) and reverse the sign: \(h>6\).
After 6 hours the temperature is exactly \(-8^{\circ}\text{F}\), so it is below \(-8^{\circ}\text{F}\) after more than 6 hours.
Answer: \(h>6\)
19. Drone descent
Inequality: \(120-15s\ge 30\).
Subtract 120: \(-15s\ge -90\).
Divide by \(-15\) and reverse the sign: \(s\le 6\).
At 6 seconds the height is exactly \(120-15(6)=30\text{ ft}\), which is allowed. Time cannot be negative, so \(0\le s\le 6\).
Answer: \(s\le 6\) seconds
20. Bake sale profit
Profit is income minus cost: \(2.5b-20\ge 85\).
Add 20: \(2.5b\ge 105\). Divide by 2.5: \(b\ge 42\).
Check: \(2.5(42)-20=105-20=85\), exactly the goal.
Answer: at least 42 boxes.
Which solution sets open the key?
Problem 1 gives x > 4 and an open dot, while Problem 2 gives x > 5. Then Problem 5 gives x ≥ 4 with a closed circle, and Problem 6 gives x ≥ -3. Finally, Problem 7 gives x ≤ 4, while Problem 8 gives x ≤ 5, so 3 and 5 pass the test and 6 fails.
Where are the flipped signs marked in the grade 7 two worksheet answer key?
In Problem 12, dividing negative 4x > 16 by negative 4 gives x < -4. In Problem 13, negative 3x ≥ 9 becomes x ≤ -3. Problem 14 gives x ≥ -10. Every flipped step carries a note, so you can compare the moment of the flip with your own work.
What are the fraction and word problem answers?
The fraction problems give x > 8 and x ≤ 15. For the school store, Jamal can buy at most 4 T-shirts, because 38 divided by 9 is about 4.2 and whole shirts only count. The same idea applies to every story problem. If the answer must be a whole number, round in the direction that keeps the inequality true.
Which slips show up most?
For example, some students flip the sign on a positive division. Others forget to flip on a negative one, and a third group draws a closed circle for a strict sign. A test value catches all three.
Turn the key into a study plan
Mark each miss as flip error, circle error, or rounding in context. Next, rework a single problem from each pile on a fresh sheet, and graph it. A few days later, try the quiz and watch whether the flips are now automatic.
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Grade 7 Two-Step Inequalities worksheet answer key
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