Solving Quadratic Inequalities · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Solving Quadratic Inequalities

Mathematics · Algebra · ages 15-16
Name ______________________   Date ____________

  1. The solution to x squared - 9 < 0 is -3 < x < 3.

    Circle one:   True   False

  2. Solve x squared + 2x - 8 > 0. What is the answer?

    • x is between -4 and 2
    • x is less than 2 or greater than -4
    • x is less than -4 or greater than 2
    • x is between -2 and 4

  3. Solve the inequality x^2 - 5x + 6 < 0.

    • 2 < x < 3
    • x < 2 or x > 3
    • x < -2 or x > -3
    • -3 < x < -2

  4. What is the first move when solving x squared - 5x + 4 > 0?

    • Set up the equation x squared - 5x + 4 = 0 and find its roots
    • Divide both sides by x squared
    • Take the square root of both sides
    • Rearrange to make x the subject

  5. Solve the inequality x^2 - 2x - 8 < 0.

    • -2 < x < 4
    • x < -2 or x > 4
    • 2 < x < 4
    • -4 < x < 2

  6. Solve the inequality x^2 - 9 >= 0.

    • x <= -3 or x >= 3
    • x < -3 or x > 3
    • -3 <= x <= 3
    • x <= -3

  7. Dev solves x squared - 5x + 6 > 0. He finds the roots 2 and 3, then writes the answer as x is between 2 and 3. Where is the slip?

    • He picked the side below the axis: a U shape above the axis is outside the roots
    • The roots should be 2 and -3
    • He should have divided both sides by x squared first
    • The endpoints should be included

  8. Solve -x squared + 4x - 3 < 0. What is the answer?

    • x is between 1 and 3
    • x is less than -1 or greater than 3
    • x is between -3 and -1
    • x is less than 1 or greater than 3

  9. How many whole numbers are in the solution set of 3x squared - 2x - 1 < 0?

    • 0
    • 2
    • 3
    • 1

  10. The quadratic x^2 - 7x + 10 has roots 2 and 5. What is the smallest integer value of x that satisfies x^2 - 7x + 10 < 0?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Solving Quadratic Inequalities W1-mt_dukJtB_0i8-s1

  1. True · The roots are -3 and 3, the curve opens up, and below the axis is the stretch between the roots.
  2. x is less than -4 or greater than 2 · The roots are -4 and 2, the curve opens up, and the middle test lands below the axis, so above the axis is the two outer stretches.
  3. 2 < x < 3 · The parabola opens upward, so it dips below the x-axis only between its two roots.
  4. Set up the equation x squared - 5x + 4 = 0 and find its roots · The balancing moves do not carry over to a quadratic inequality, so you set the equals version and find the roots. They are the boundaries of the answer.
  5. -2 < x < 4 · The upward parabola dips below the axis only in the strip between its two roots.
  6. x <= -3 or x >= 3 · Because the sign includes 'equal to', the roots themselves belong to the solution set.
  7. He picked the side below the axis: a U shape above the axis is outside the roots · For a U shape, the stretch between the roots is where the curve is below the axis. The inequality wants above the axis, so the answer is the two outer stretches.
  8. x is less than 1 or greater than 3 · Multiplying by -1 flips the sign to x squared - 4x + 3 > 0. The roots are 1 and 3, and a U shape above the axis is outside the roots.
  9. 1 · The factors are (3x + 1)(x - 1), so the roots are -1/3 and 1. Below the axis is between them, and the only whole number between -1/3 and 1 is 0.
  10. 3 · The inequality holds strictly between the roots, so the smallest whole number greater than 2 is 3.
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