➕ Algebra 1 · Inequalities

Solving Inequalities practice questions

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Answer key with explanations

1. Solve x + 8 < 3.

   Answer: x < −5. Subtract 8 from both sides. The sign stays: x < −5. Check: x = −6 gives 2 < 3 (true), but x = −5 gives 3 < 3 (false), so −5 is not included. The choice x < 11 comes from adding 8 instead of subtracting it.

2. A bus holds no more than 36 passengers. Let p be the number of passengers. Which inequality describes this?

   Answer: p ≤ 36. “No more than 36” means 36 or fewer. A full bus (exactly 36) is allowed, so use ≤, not <. The choice p ≥ 36 means full or over-full, and p > 36 means over-full.

3. A number line shows a closed circle at −2 with the shading going to the right. Which inequality does it show?

   Answer: x ≥ −2. A closed circle means the boundary −2 is included, so the sign is ≥ or ≤. Shading to the right covers the numbers greater than −2, so the inequality is x ≥ −2. Test: x = −2 works, x = 0 works, and x = −3 does not.

4. Solve −5x ≤ 35.

   Answer: x ≥ −7. Divide both sides by −5. Dividing by a negative number reverses the sign: x ≥ −7. Check: x = 0 gives 0 ≤ 35 (true), and x = −10 gives 50 ≤ 35 (false). If you forget to reverse the sign you get x ≤ −7, which would include −10.

5. Which number is a solution of −2x + 5 < −9?

   Answer: 8. Subtract 5: −2x < −14. Divide by −2 and reverse the sign: x > 7. So 8 works: −2(8) + 5 = −11, and −11 < −9 is true. The number 7 is the boundary: −2(7) + 5 = −9, and −9 < −9 is false. The numbers 0 and −7 are less than 7, so they fail: −2(0) + 5 = 5 and −2(−7) + 5 = 19, and neither is less than −9.

6. Solve 2(3x + 1) > 5x + 13.

   Answer: x > 11. Distribute: 6x + 2 > 5x + 13. Subtract 5x: x + 2 > 13. Subtract 2: x > 11. Check: x = 12 gives 74 > 73 (true), but x = 11 gives 68 > 68 (false). If you forget to distribute the 2 onto the +1, you get x > 12.

7. A taxi charges a 4-dollar base fee plus 2.50 dollars per mile. Dana wants to spend at most 20 dollars. Let m be the number of miles. Which inequality describes the trips she can take?

   Answer: 2.5m + 4 ≤ 20. The cost is the base fee plus the per-mile charge times the miles: 2.5m + 4 dollars. “At most 20” means the cost is ≤ 20. The other choices have the wrong sign or put the fee and the rate in the wrong places. For example, a 6-mile trip costs 2.5(6) + 4 = 19 dollars, which is allowed.

8. Solve 9 − 3x ≥ 5x − 23.

   Answer: x ≤ 4. Add 3x to both sides to keep the variable on the right, where its coefficient is positive: 9 ≥ 8x − 23. Add 23: 32 ≥ 8x. Divide by 8: 4 ≥ x, which is x ≤ 4. Check: x = 4 gives −3 ≥ −3 (true), x = 0 gives 9 ≥ −23 (true), and x = 5 gives −6 ≥ 2 (false).

9. Solve (−2/5)x + 3 < 7.

   Answer: x > −10. Subtract 3: (−2/5)x < 4. Multiply both sides by −5/2, which is negative, so reverse the sign: x > −10. Check: x = −10 gives 4 + 3 = 7, and 7 < 7 is false, so −10 is not included. x = 0 gives 3 < 7 (true). x = −15 gives 6 + 3 = 9, and 9 < 7 is false. If you forget to reverse the sign you get x < −10.

10. Which statement describes the solutions of 5(x − 1) + 3 ≥ 5x − 2?

    Answer: Every real number is a solution.. Distribute: 5x − 5 + 3 ≥ 5x − 2, so the left side simplifies to 5x − 2. The inequality is 5x − 2 ≥ 5x − 2. Subtract 5x from both sides: −2 ≥ −2. That is true (the two sides are equal, and ≥ allows equal), so every real number is a solution. With > instead of ≥, there would be no solution.

11. A hall costs 35 dollars to rent plus 6 dollars per guest for food. A club has at most 100 dollars to spend. What is the greatest whole number of guests it can have?

    Answer: 10 guests. Let g be the number of guests. The cost is 6g + 35 ≤ 100. Subtract 35: 6g ≤ 65. Divide by 6: g ≤ 65/6, which is between 10 and 11 (about 10.8). It is closer to 11, but 11 does not satisfy the inequality. Guests are whole numbers, and the solved inequality g ≤ 65/6 says g can be at most 65/6, so the greatest whole number that works is 10 (round down). Check: 10 guests cost 6(10) + 35 = 95 dollars. 11 guests cost 101 dollars, which is too much. The choices 16 and 17 come from forgetting the 35-dollar rental fee.