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Inequalities

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Lesson notes

Inequality symbols

  • < less than, > greater than, \leq less than or equal to, \geq greater than or equal to.
  • -2 < x \leq 3 means x is greater than -2 and at most 3. Integers satisfying it: -1, 0, 1, 2, 3.

Solving linear inequalities

  • Solve like an equation, doing the same to both sides. The answer is a range: 3x + 2 > 11 gives x > 3.
  • Reverse the inequality sign if you multiply or divide by a negative number: -2x < 8 gives x > -4.

Number lines and set notation

  • A filled circle shows the end value is included (\leq, \geq); an open circle shows it is excluded (<, >).
  • Write a double inequality such as -1 \leq x < 4 as one range on a number line.

Quadratic inequalities

  • Solve x2 - x - 6 < 0: factorise (x - 3)(x + 2) < 0. Roots -2 and 3. The parabola is below the axis between the roots, so -2 < x < 3.
  • For x2 > 9: x < -3 or x > 3.

Where the quadratic is negative

Where the quadratic is negativeO−8−6−4−22468101214−4−3−2−112345xy(−2, 0)(3, 0)y = x² −x − 6

Graphical inequalities

  • Draw a solid boundary for ≤ or ≥, and a dashed boundary for < or >. Test a point not on the boundary to choose the required side; the origin works only if it is off the boundary.
  • Regions defined by several inequalities: shade out the unwanted regions and leave the required region unshaded, or as the question states.

Slides

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Practice questions

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  1. 1.Which symbol means "less than or equal to"?

    Easy
    • A<
    • B\leq
    • C>
    • D\geq

  2. 2.Solve x + 4 < 10.

    Easy
    • Ax < 6
    • Bx > 6
    • Cx < 14
    • Dx > 14

  3. 3.Solve 2x \geq 8.

    Easy
    • Ax \geq 4
    • Bx \leq 4
    • Cx \geq 16
    • Dx \geq 6

  4. 4.Which integers satisfy -2 < x \leq 1?

    Easy
    • A-1, 0, 1
    • B-2, -1, 0, 1
    • C-1, 0
    • D-2, -1, 0

  5. 5.Solve 3x - 5 \leq 10.

    Easy
    • Ax \leq 5
    • Bx \geq 5
    • Cx \leq 15
    • Dx \leq 1.67

  6. 6.Solve -2x < 8.

    Easy
    • Ax < -4
    • Bx > -4
    • Cx < 4
    • Dx > 4

  7. 7.Solve 5 - x > 2.

    Easy
    • Ax > 3
    • Bx < 3
    • Cx > -3
    • Dx < -3

  8. 8.What does an open circle on a number line mean?

    Easy
    • Athe value is included
    • Bthe value is not included
    • Cthe value is negative
    • Dthere is no solution

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