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Grade 7 · unit review

Review: Equations and Inequalities with Rational Structure

Solving means finding inputs that make a relationship true. Balanced operations preserve equality, whereas inequality direction must reverse when multiplication or division by a negative number reverses number-line order.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Solve x/2+8=15.
Need a hint?
  • A coefficient can be a fraction, such as one half.
  • Perform the same operation on both sides, including the constant.
2. Solve 7(x−1)=49.
Need a hint?
  • A nonzero factor multiplying an entire group can be divided out before expanding.
  • Division must affect the entire side of the equation.
3. Solve −2x < -14.
Need a hint?
  • Multiplying an ordered pair of numbers by a negative reflects their positions across zero.
  • Reverse the sign only when multiplying or dividing both sides by a negative.
4. A fictional model has a fixed cost $7 plus $5 per item x. Spending must be at most $42. Which whole-number counts are allowed?
Need a hint?
  • An at-most constraint includes its boundary and can be expressed with ≤.
  • A fraction may solve the algebraic inequality but cannot represent a count of whole items.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Solve x/2+10=19. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. 2. Solve 9(x−1)=81. Candidate answers: (A) 10 (B) 11 (C) 20 (D) 9. 3. Solve −2x < -18. Candidate answers: (A) x < 9 (B) x = 9 (C) x > -9 (D) x > 9. 4. A fictional model has a fixed cost $7 plus $5 per item x. Spending must be at most $52. Which whole-number counts are allowed? Candidate answers: (A) 0 through 10 (B) 0 through 8 (C) 0 through 9 (D) 9 or more.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 18. Subtract 10 from both sides to get x/2=9. Multiply both sides by 2: x=18. Check: 18/2+10=19.0. 2. 10. Divide both sides by 9: x−1=9. Add 1 to both sides: x=10. Substitution gives 9(10−1)=81. 3. x > 9. Dividing both sides by −2 reverses the inequality direction. The boundary is -18/(−2)=9; equality is excluded. 4. 0 through 9. Write 5x+7≤52. Subtract 7 and divide by 5 to obtain x≤9. A count must also be a nonnegative integer.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.