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

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)

⇒ |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.

⇒ |-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

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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, ∞).

E**xample 6:** Solve the inequality (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).