### Shortcut Methods

**Typical numericals on Linear Inequality for JEE and CBSE board exams:**

**1.** Solve the inequality: 2x + 5 < 15.
**Solution**: 2x < 10
**Therefore, x < 5.**

**2.** Find the solution set of the inequality: -3x + 4 > -5.
**Solution**: -3x > -9
**Therefore, x < 3.**

**3.** Solve the inequality: 4 - 2x ≥ 6.
**Solution**: -2x ≥ 2
**Therefore, x ≤ -1.**

**4.** Find the interval of values of x for which the inequality 3x - 5 < 10 - 2x holds true.
**Solution**: 3x - 5 < 10 - 2x
**5x < 15**
**Therefore, x < 3.**

**5.** Determine the range of values of x for which the inequality 2x + 3 ≤ 5x - 2 is satisfied.
**Solution**: 2x + 3 ≤ 5x - 2
**3x ≤ -5**
**Therefore, x ≤ -5/3.**

**6.** Solve the inequality: 2(3x - 4) > 5(2x + 1) - 3.
**Solution**: 6x - 8 > 10x + 5 - 3
**6x - 8 > 10x - 2**
**10x - 6x > 2+8**
**4x > 10**
**Therefore, x > 5/2.**

**7.** Find the solution set of the inequality: 4 - 3(2x + 5) ≥ 7 - 2(3x - 4).
**Solution**: 4 - 6x - 15 ≥ 7 - 6x + 8
**-11 - 6x ≥ 15 - 6x**
**-11 ≥ 15**
**Therefore, no real solution exists.**

**8.** Determine the range of values of x for which the inequality (x + 2)/(x - 1) < 2 is true.
**Solution**: (x + 2)/(x - 1) < 2
**x + 2 < 2(x - 1)**
**x + 2 < 2x - 2**
**x - 2x < -2 -2**
**-x < -4**
**Therefore, x > 4.**

**9.** Solve the inequality: 1/3(x - 2) > 1/2(x + 1) + 5.
**Solution**: 1/3(x - 2) > 1/2(x + 1) + 5
**x - 2 > 3/2(x + 1) + 15**
**x - 2 > 3/2x + 3/2 + 15**
**2(x - 2) > 3(3/2x + 3/2 + 15)**
**2x - 4 > 9/2x + 9/2 + 45**
**2x - 9/2x > 9/2 + 45 + 4**
**5/2x > 59/2**
**Therefore, x > 59/5.**

**10.** Find the solution set of the inequality: (2x - 3)/(4 - x) ≥ 1.
**Solution**: (2x - 3)/(4 - x) ≥ 1
**(2x - 3) ≥ 1(4 - x)**
**2x - 3 ≥ 4 - x**
**2x + x ≥ 4 + 3**
**3x ≥ 7**
**Therefore, x ≥ 7/3.**