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Math: Intermediate Algebra for the ACT Exam

Intermediate Algebra builds directly on Pre-Algebra and is the most heavily tested Algebra sub-area on the ACT. High-yield topics include: solving and graphing linear inequalities, working with systems of two equations, solving quadratics by factoring or the quadratic formula, evaluating function notation (f(x)), and working with absolute value equations. These question types appear consistently across every ACT administration and reward systematic algebra skills over creative problem-solving.

ACT Inc. — ACT Test Specifications: Mathematics section content areas and question distribution.

Locale-specific study guides

Pass-rate data, regulatory context, and study tips for Math: Intermediate Algebra all change by candidate locale. Pick your context:

Common failure modes

These are the patterns that cause most candidates to lose marks on this topic. Recognising them in advance is half the work.

  • !Forgetting to flip the inequality sign when multiplying or dividing by a negative
  • !Solving only one equation in a system problem and selecting a partial answer ("x = 3" when asked for "x + y")
  • !Missing extraneous solutions in absolute value equations — always check both solutions by substituting back
  • !Confusing function composition f(g(x)) with multiplication f(x) × g(x)

Study tips

  • 1Solve 10 systems-of-equations problems using both substitution and elimination methods. Know which method is faster for different configurations (e.g., elimination when coefficients differ by a multiple).
  • 2For quadratic questions, check whether the expression factors easily before resorting to the quadratic formula. Factorable quadratics take under 30 seconds; quadratic formula takes 60+ seconds.
  • 3Absolute value equations always produce two equations: |x − 3| = 7 means x − 3 = 7 or x − 3 = −7. Check both solutions in the original equation.
  • 4Practice function notation: f(3) means evaluate the function at x = 3, not multiply f times 3. Practice composite functions f(g(2)) by evaluating from the inside out.

Sample ACT Math: Intermediate Algebra questions

These sample items mirror the format and difficulty of real ACT questions. Practice with thousands more on the free Koydo question bank.

  1. 1

    Solve for x and y: 2x + y = 10 and x − y = 2.

    • Ax = 3, y = 4
    • Bx = 4, y = 2Correct
    • Cx = 4, y = 6
    • Dx = 6, y = −2
    Why this answer?

    Adding the equations: (2x + y) + (x − y) = 10 + 2 → 3x = 12 → x = 4. Substituting x = 4 into x − y = 2: 4 − y = 2 → y = 2. Check: 2(4) + 2 = 10 ✓ and 4 − 2 = 2 ✓. (Illustrative.)

  2. 2

    For what values of x is |2x − 6| < 4?

    • A1 < x < 5Correct
    • Bx < 1 or x > 5
    • C−1 < x < 5
    • Dx > 5
    Why this answer?

    |2x − 6| < 4 means −4 < 2x − 6 < 4. Add 6 to all parts: 2 < 2x < 10. Divide by 2: 1 < x < 5. This is the absolute-value inequality method: rewrite as a compound inequality and solve both inequalities simultaneously.

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