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Math: Algebra, 3 original questions

SAT Linear Inequalities Practice: Questions and Answers

Work through original SAT-style inequality questions about negative division, feasible coordinate pairs and whole-number budgets, with complete explanations.

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The short answer

A linear inequality describes a range of permitted values rather than requiring exact equality. Simplify it much like an equation, but reverse the inequality sign when multiplying or dividing both sides by a negative number. For a two-variable inequality, substitute a coordinate pair to test whether it meets the condition. In context problems, decide whether the endpoint is included and whether the variable must be a whole number. Check a permitted value and interpret the result using the stated constraints.

Method at a glance

  1. 1Translate the condition
  2. 2Isolate the variable carefully
  3. 3Test a boundary or point
  4. 4Apply whole-number constraints

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Translate the condition

    At most uses ≤, at least uses ≥, and strictly more or less excludes equality. Include all stated context restrictions.

  2. Step 2: Isolate the variable carefully

    Apply balanced operations. Reverse the comparison when multiplying or dividing by a negative quantity.

  3. Step 3: Test a boundary or point

    Substitute the endpoint or a candidate coordinate pair into the original inequality, including equality when allowed.

  4. Step 4: Apply whole-number constraints

    If the unknown counts objects or sessions, choose an integer that meets the bound. Verify the proposed maximum or minimum.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

Which inequality is equivalent to −3x + 5 > 20?

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Correct answer

Choice B: x < −5

Explanation

Subtract 5 to get −3x > 15. Divide by −3 and reverse the sign, giving x < −5. For example, x = −6 makes the original left side 23, which is greater than 20.

Why each choice is right or wrong

  1. Choice A

    This fails to reverse the inequality after negative division.

  2. Choice B

    Correct answer

    Correct: dividing by −3 gives the endpoint −5 and reverses the comparison.

  3. Choice C

    The endpoint is −5, not 5.

  4. Choice D

    Both the endpoint sign and the comparison direction are incorrect.

Question 2 of 3

Which coordinate pair satisfies 2x + y ≤ 9?

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Correct answer

Choice C: (2, 4)

Explanation

For (2, 4), the left side is 2 × 2 + 4 = 8, which is at most 9. The other pairs each give 10 and therefore fail the inequality.

Why each choice is right or wrong

  1. Choice A

    2 × 4 + 2 = 10, which is greater than 9.

  2. Choice B

    2 × 3 + 4 = 10, so this pair is not permitted.

  3. Choice C

    Correct answer

    Correct: this pair gives 8, which satisfies the upper bound of 9.

  4. Choice D

    2 × 5 + 0 = 10, which exceeds the bound.

Question 3 of 3

A club pays a fixed $18 charge plus $4 for each notebook. Its budget is at most $50. What is the greatest whole number of notebooks it can buy?

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Correct answer

Choice B: 8

Explanation

If n is the number of notebooks, 18 + 4n ≤ 50. Subtract 18 to get 4n ≤ 32, so n ≤ 8. Eight notebooks cost exactly $50 and are allowed; nine cost $54 and are not.

Why each choice is right or wrong

  1. Choice A

    Seven are affordable, but eight also fit the budget, so seven is not the greatest number.

  2. Choice B

    Correct answer

    Correct: eight notebooks meet the budget exactly.

  3. Choice C

    Twelve notebooks cost $66 including the fixed charge.

  4. Choice D

    Seventeen notebooks cost $86 including the fixed charge.

Common mistakes

  • Dividing by a negative number without reversing the comparison. Test a simple value if the direction feels uncertain.

  • Replacing at most with a strict less-than sign. The maximum itself is allowed when the wording includes equality.

  • Rounding a maximum count upward past the budget. The chosen whole number must still satisfy the original inequality.

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Common questions

When do I reverse an inequality sign?

When multiplying or dividing both sides by a negative number. Adding or subtracting a quantity does not itself reverse the sign.

How can I check a two-variable solution?

Substitute the pair into the original inequality. The comparison must be true, with equality accepted only when the symbol permits it.

Do I include the boundary value?

Use the wording and symbol. At most and at least include it; strictly less or greater exclude it.

Sources

  1. College Board: Algebrasatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides