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Drill by score band: 400–500 500–600 600–700 700–800

Linear Inequalities · Sub-skill drill

Compound and Absolute Value Inequalities

Compound inequalities combine two inequalities joined by 'and' or 'or'. The 'and' case represents an intersection on the number line and is usually written as a single chained inequality like a < x < b. The 'or' case represents a union and is usually written as two separate inequalities. Absolute value inequalities reduce to one of these two cases: |x| < a becomes a chained inequality, while |x| > a becomes an or-statement. The wrong-answer pattern is almost always swapping the chained form for the or-form or vice versa.

How this sub-skill is tested on the SAT

Compound inequalities combine two inequalities joined by 'and' or 'or'. The 'and' case represents an intersection on the number line and is usually written as a single chained inequality like a < x < b. The 'or' case represents a union and is usually written as two separate inequalities. Absolute value inequalities reduce to one of these two cases: |x| < a becomes a chained inequality, while |x| > a becomes an or-statement. The wrong-answer pattern is almost always swapping the chained form for the or-form or vice versa.

This sub-skill sits inside the broader Linear Inequalities topic, which is part of the College Board's Heart of Algebra content domain. Heart of Algebra accounts for roughly a third of every SAT Math section. The College Board frames it as the ability to analyze, fluently solve, and create linear equations and inequalities — and to interpret what their solutions mean in context. If you walk into test day weak here, no amount of advanced math fluency will compensate, because Heart of Algebra questions appear in both the calculator and no-calculator modules and are usually front-loaded so they set the tempo for the entire section. Mastery looks like solving a two-step linear equation in under twenty seconds, recognizing parall

Practice questions in this drill set

Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.

  1. 400-500 easy

    Which of the following describes all values of x for which 8x + 7 > 7?

    1. A x < 0
    2. B x > 0
    3. C x <= -1
    4. D x >= 1
    Worked solution

    Answer: B — x > 0

    Subtract 7 from both sides: 8x > 0. Divide both sides by 8: x > 0.

    Open question q0027 in its own page →

  2. 400-500 easy

    Which of the following describes all values of x for which 7x + 1 > 15?

    1. A x < 2
    2. B x <= 1
    3. C x >= 3
    4. D x > 2
    Worked solution

    Answer: D — x > 2

    Subtract 1 from both sides: 7x > 14. Divide both sides by 7: x > 2.

    Open question q0031 in its own page →

  3. 500-600 medium

    Which of the following describes all values of x for which -8x - 2 > -10?

    1. A x > -1
    2. B x < -1
    3. C x <= -2
    4. D x >= 0
    Worked solution

    Answer: B — x < -1

    Subtract -2 from both sides: -8x > -8. Divide both sides by -8 (negative, so flip the inequality): x < -1.

    Open question q0035 in its own page →

  4. 600-700 medium

    Which of the following describes all values of x for which 2x - 8 > -18?

    1. A x > -5
    2. B x >= -4
    3. C x < -5
    4. D x <= -6
    Worked solution

    Answer: A — x > -5

    Subtract -8 from both sides: 2x > -10. Divide both sides by 2: x > -5.

    Open question q0039 in its own page →

  5. 600-700 medium

    Which of the following describes all values of x for which 7x - 8 > -15?

    1. A x > -1
    2. B x < -1
    3. C x <= -2
    4. D x >= 0
    Worked solution

    Answer: A — x > -1

    Subtract -8 from both sides: 7x > -7. Divide both sides by 7: x > -1.

    Open question q0043 in its own page →

  6. 700-800 hard

    Which of the following describes all values of x for which -7x + 3 > 3?

    1. A x < 0
    2. B x > 0
    3. C x <= -1
    4. D x >= 1
    Worked solution

    Answer: A — x < 0

    Subtract 3 from both sides: -7x > 0. Divide both sides by -7 (negative, so flip the inequality): x < 0.

    Open question q0047 in its own page →

Why this band assignment matters

Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.

Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.

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