20. Solving Other Inequalities

Notes:

Solving an inequality that involves a quadratic expression

To solve a linear inequality, we isolate x on one side of the inequality with just numbers on the other side. For example, 7x – 4 < 4x + 5 simplifies to x < 3, i.e., x ∈ (-∞, 3]. What about an inequality that involves a quadratic expression such as x² + 4x – 3 ≤ 5x + 9?

• x² + 4x – 3 – 5x – 9 ≤ 0
• x² – x – 12 ≤ 0
• (x + 3)(x – 4) ≤ 0

Since the quadratic on the left-hand side factors, we know that the roots are x = -3 and x = 4 and since the coefficient on x² is positive we know the parabola opens upwards. Therefore, the solution must be -3 ≤ x ≤ 4, i.e., x ∈ [-3, 4].

We can check with test values:

• (-4)² + 4(-4) – 3 ≤ 5(-4) + 9 ⇒ -3 ≤ -11: false
• 1² + 4(1) – 3 ≤ 5(1) + 9 ⇒ 2 ≤ 14: true
• 5² + 4(5) – 3 ≤ 5(5) + 9 ⇒ 42 ≤ 34: false

Solving an inequality that involves a rational expression

What about an inequality that involves a rational expression such as (x + 2) / (x – 1) ≥ 0?

• When x = 1, (x + 2) / (x – 1) is undefined.
• When x = -2, y = 0.
• When x < -2, numerator is negative, denominator is negative, so y > 0.
• When -2 < x < 1, numerator is positive, denominator is negative, so y < 0.
• When x > 1, numerator is positive, denominator is positive, so y > 0.

So, the solution is x ≤ -2 or x > 1, i.e., x ∈ (-∞, -2] ∪ (1, ∞).

Solving an inequality that involves an absolute value

What about an inequality that involves an absolute value such as |2x – 3| < 5?

• -5 < 2x – 3 < 5

• -1 < x < 4, i.e., x ∈ (-1, 4)
• Check: |2(1) – 3| < 5, |-1| < 5, 1 < 5

Another example: |4x – 6| > 2?

• 4x – 6 < -2 or 4x – 6 > 2

• x < 1 or x > 2, i.e., x ∈ (-∞,1) ∪ (2,∞)
• Check: |4(-1) – 6| > 2, 10 > 2 and |4(3) – 6| > 2, 6 > 2

Other examples

Find the values of x such that 5x² + 2x – 5 > x² + 2x + 4.

• 5x² + 2x – 5 – x² – 2x – 4 > 0
• 4x² – 9 > 0
• (2x + 3)(2x – 3) > 0
• Solution: x < -3/2 or x > 3/2, i.e., x ∈ (-∞, -3/2) ∪ (3/2, ∞).
• Check: 5(-2)² + 2(-2) – 5 > (-2)² + 2(-2) + 4, 11 > 4.
• Check: 5(2)² + 2(2) – 5 > (2)² + 2(2) + 4, 19 > 12.

Find the values of x such that |-3x + 6| ≤ 9.

• -9 ≤ -3x + 6 ≤ 9
• -9 – 6 ≤ -3x ≤ 9 – 6

• 5 ≥ x ≥ -1
• Solution: -1 ≤ x ≤ 5, i.e., x ∈ [-1,5].
• Check: |-3(1) + 6| ≤ 9, 3 ≤ 9.