Name: ___________________________ Date: ______________ Class: ____________
LEVEL 2 · ENRICHMENT

Equations & Inequalities

Standards: 6.EE.B.5–8 (Solving Equations, Writing Equations, Inequalities)

Directions: Solve each problem and show all steps. Write equations or inequalities before solving word problems. For "Explain" prompts, justify in complete sentences. Finish with the Stretch Problem.

Challenge Problems

1. Solve and check: 3x + 7 = 28 MULTI-STEP

2. Write an equation and solve: a number divided by 5, then increased by 4, equals 10. TRANSLATE

3. Find all whole-number solutions to x + 3 < 10 where x ≥ 0. REASONING

4. Two consecutive even numbers have a sum of 54. Write an equation and find both numbers. REAL-WORLD

5. A rectangle has a perimeter of 38 cm. Its length is 3 cm more than its width. Find the dimensions. MULTI-STEP

6. Marcus has $45. He buys n notebooks at $3.50 each and has at least $10 left. Write and solve an inequality. REAL-WORLD

7. Create a real-world situation whose solution is the equation 2x + 5 = 23. Solve it and explain your context. OPEN-ENDED

8. For what values of x are BOTH x + 2 > 5 AND 2x < 18 true at the same time? REASONING

9. A triangle has angles x°, (x + 20)°, and (2x − 10)°. Find all three angle measures. MULTI-STEP

Remember the angles of a triangle sum to 180°.

10. Phone Plan A costs $25/month plus $0.10 per text. Plan B costs $35/month, unlimited texts. Write an inequality to find when Plan A is cheaper. REAL-WORLD

Explain Your Reasoning

11. A student solved 5x = 30 and wrote x = 35. What mistake was made, and what is the correct solution? Explain how to check an answer. WRITE

★ Stretch Problem

A school is renting a bus for $180 plus $4 per student. The budget cannot exceed $320. (a) Write an inequality for the number of students s. (b) What is the greatest number of students who can attend? (c) Explain what happens to the cost-per-student if exactly 35 students attend.

Answer Key

  1. x = 7 — 3x = 21; check 3(7)+7 = 28 ✓
  2. x/5 + 4 = 10 → x/5 = 6 → x = 30
  3. x < 7, so {0,1,2,3,4,5,6} — seven solutions
  4. 2x + 2 = 54 → x = 26; numbers are 26 and 28
  5. 2w + 2(w+3) = 38 → 4w = 32 → w = 8; width 8 cm, length 11 cm
  6. 45 − 3.50n ≥ 10 → n ≤ 10; at most 10 notebooks
  7. Sample: $5 entry + $2/hour, total $23. 2x + 5 = 23 → x = 9 hours
  8. x > 3 AND x < 9, so 3 < x < 9
  9. 4x + 10 = 180 → x = 42.5; angles 42.5°, 62.5°, 75°
  10. 25 + 0.10t < 35 → t < 100; cheaper when fewer than 100 texts
  11. Divided wrong / divided 30 by... should divide by 5: x = 6. Check: 5(6) = 30 ✓ (5×35 = 175, not 30)
  12. Stretch: (a) 180 + 4s ≤ 320; (b) 4s ≤ 140 → s ≤ 35, so 35 students; (c) at 35 students total = $320, so $320/35 ≈ $9.14 per student