Inequalities: Solved Examples

Example 1: Solve the inequality 2x+1/x+3 < 1
Solution: We have 2x+1/x+3< 1
Here the right hand side is not equal to zero. So transpose 1 to the left hand side and solve. We get 2x+1/x+3 < 1
⇒ (2x+1/x+3)-1 < 0
⇒ (2x+1-x-3/x+3) <0
⇒ (x-2/x+3) < 0
The critical points are x = 2, -3. Plot these points on the number line, we get

Since the given inequality is negative, so the solution is -3 < x < 2
Example 2: Solve the inequality x-4/2x-1 > 2
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Solution: We have x-4/2x-1 > 2
Here the right hand side is not equal to zero. So transpose 1 to the left hand side and solve. We get (x-4/2x-1) - 2 > 0
⇒ x-4-4x+2/1x-1 > 0
⇒ -3x-2/2x-1 >0
⇒ 3x+2/2x-1 >0
The critical points are x = (-2/3), (1/2)

The solutions is (-2/3) < x < (1/2)(1/2)
Example 3: Solve |9x – 4| - 5 < 11.
Solution: We have |9x – 4| - 5 < 11
⇒ |9x – 4| < 11 + 5
⇒ |9x – 4| < 16
⇒ - 16 < 9x – 4 < 16
⇒ - 16 + 4 < 9x < 16 + 4
⇒ -12 < 9x < 20 ⇒ (-12/9) < x < (20/9)
⇒ (-4/3) < x < (20/9) , which is the required solution.
Example 4: Solve |-3x + 7| + 8 < 15.
Solution: We have |-3x + 7| + 8 < 15
⇒ |-3x + 7| < 15 – 8
⇒ |-3x + 7| < 7
⇒ - 7 < - 3x + 7 < 7 ⇒ - 7 – 7 < -3x < 0
⇒ - 14 < - 3x < 0 ⇒ 0 < 3x < 14
⇒ 0 < x < 14/3, which is the required solution
Example 5: Solve (x – 7)2 (2x + 5) (x + 7)3 > 0.
Solutions: We have (x – 7)2 (2x + 5) (x + 7)3 > 0
The critical points are x = -7, (-5/2),7
Plot these points on the number line and draw curve

Since our inequality is positive, the solution is x ε (-∞, -7) ∪ ( (-5/2),7) ∪ (7, ∞).
Example 6: Solve the inequality (3x-7)2(x+6)/(x-5)4 ≤ 0
Solutions: We have (3x-7)2(x+6)/(x-5)4 ≤ 0
The critical points are x = -6,7/3,5
Plot these points on the number line and draw the curve

Since the given inequality is negative, so the solution is x ε (-∞, -6).
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