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Grade 9 · unit review

Review: Algebra1: Solution Sets and Parameters

An equation may describe distance, a bounded interval, a parameter-dependent relation or a formula between several quantities. Its solution must respect all stated conditions, including endpoint inclusion and nonzero divisors.

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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−7|=8.
Need a hint?
  • Absolute value measures nonnegative distance from zero.
  • Do not keep only the positive direction.
2. Solve 15≤2x+1<21.
Need a hint?
  • A compound inequality restricts one expression between two bounds.
  • An inclusive left boundary and strict right boundary need separate symbols.
3. For what real k does kx+7=9x+8 have no solution?
Need a hint?
  • A parameter controls an equation’s structure.
  • Do not confuse a parameter value with a variable solution.
4. Rearrange V=πr²h to solve for h, with r≠0.
Need a hint?
  • Rearranging a formula uses the same balanced operations as solving a numerical equation.
  • Parentheses determine which entire group is multiplied.

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−9|=10. Candidate answers: (A) No solution because an absolute value is positive (B) x=-1 or x=19 (C) x=1 only (D) x=19 only. 2. Solve 19≤2x+1<25. Candidate answers: (A) 18≤x<24 (B) 9<x≤12 (C) 9≤x<12 (D) 9≤x≤12. 3. For what real k does kx+9=11x+10 have no solution? Candidate answers: (A) 0 (B) 10 (C) 11 (D) 12. 4. Rearrange y=kx+b to solve for b. Candidate answers: (A) b=(y−k)x (B) b=y+kx (C) b=y/k−x (D) b=y−kx.

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. x=-1 or x=19. A distance of 10 from 9 permits both directions. Set x−9=10 or x−9=−10; the solutions are 19 and -1. 2. 9≤x<12. Subtract 1 from every part, then divide every part by positive 2. The left endpoint stays inclusive and the right stays exclusive. 3. 11. At k=11, the variable coefficients cancel and leave 9=10, false. At any other k, a nonzero coefficient gives one solution. 4. b=y−kx. Subtract the entire kx term from both sides. No division is required, so k=0 remains allowed for this rearrangement.

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