Solving linear Inequalities means that comparison of two values or expressions. A linear inequality means that a relationship between two quantities that are not equal. In equations, one side is equal to the other side. To solving the linear inequalities, multiply, divide, or subtract the both side of the inequality equation to simplify the equation. Divide or multiplying inequalities by a negative number, must flip the inequality sign.**An linear inequalities which consists of more than one terms are raised only to the first power in terms, having the following forms,**

bx + c > 0

bx + c < 0

bx + c $\geq$ 0

bx + c $\leq$ 0

Where "b" represents the numerical coefficient of x, and "c" represents the constant term.

Properties Used on Solving Linear Inequalities:

Let a, b and c be real numbers.

1. **Transitive Property**

If a < b and b < c then a < c

2. **Addition Property**

If a < b then a + c < b + c

3. **Subtraction Property**

If a < b then a - c < b - c

4. **Multiplication Property**

i. If a < b and c is positive then c $\times$ a < c $\times$ b

ii. If a < b and c is negative c $\times$ a > c $\times$ b**Example 1:** Solving the linear inequalities 6x - 7 > 2x + 1**Solution:**

Given

6x - 7 > 2x + 1

Add 7 to both sides and simplifying the equation

6x > 2x + 8

Subtract 2x to both sides and simplifying the equation

4x > 8

Multiply both sides by $\frac{1}{4}$ and simplifying the equation

x > 2 **Conclusion:** The solution the interval (2, + infinity).**Example 2:** Solve the linear inequality 6x - 4 > 2x + 4**Solution:**

Add 4 to both sides and simplifying the equation

6x > 2x + 8

Subtracting 2x on both sides and simplifying the equation

4x > 8

Multiply both sides by $\frac{1}{4}$ and simplifying the equation

x > 8**Example 3:** Solve the linear inequalities 7x + 12 > 0**Solution:**

From the given problem 7x + 12 > 0

Subtract 12 on both sides, we get

7x + 12 - 12 > 0 - 12

7x > - 12

Divide by 7 on both sides,

$\frac{(7x)}{7}$ > - $\frac{12}{7}$

x > -1.71

**Example 4:** Solve the linear inequalities 3x + 20 > 0**Solution: **

From the given problem 3x + 20 > 0

Subtract 20 on both sides, we get

3x + 20 - 20 > 0 - 20

3x > - 20

Divide by 3 on both sides,

$\frac{(3x)}{3}$ > - $\frac{20}{3}$

x > - 6.33**Example 5:** Solve the linear inequalities 5x - 25 < 0**Solution:**

From the given problem 5x - 25 < 0

Add 25 on both sides, we get

5x + 25 - 25 < 0 + 25

5x < 25

Divide by 5 on both sides,

$\frac{(5x)}{5}$ < $\frac{25}{5}$

x < 5

** ****Example 6:** Solve the linear inequalities - 4x - 24 < 0

Solution:

From the given problem - 4x - 24 < 0

Add 24 on both sides, we get

- 4x + 24 - 24 < 0 + 24

- 4x < 24

Divide by -4 on both sides,

$\frac{(-4x)}{-4}$ < $\frac{24}{-4}$

When we divide the negative term the inequality sign β<β changes toβ >β

x > 6

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