Two algebraic expressions or two real numbers related by the symbol ‘<’, ‘>’, ‘<=’ or ‘>=’ form an** **linear ** **inequality.

Examples of numerical inequalities: 3 < 5; 7 > 5 x < 5;

Examples of literal inequalities: y > 2; x <= 3, y >= 4.

3 < x < 5 (read as x is greater than or equal to 3 and less than 5) and 2 < y < 4 are the examples of double inequalities.

**How to solve the linear inequality:**

Added or subtracted by the same number on both side of the inequality.

Multiplied or divided by the same number on both side of the inequality but if we divide or multiply by a negative number, we must reverse the inequality sign.

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**Example problem 1:**

Solve the linear inequality: 5*x *– 3 < 7

**Solution:**

5*x *– 3 < 7

Add 3 on both side of the equation

5x - 3 + 3 < 7 + 3

5x < 10

Divide by 5 on both side of the equation

5x / 5 < 10 / 5

x<2

So, the solution is (-infinity, 2).

**Example problem 2:**

Solve the linear inequality: 7*x *+ 3 < 5*x *+ 9

**Solution:**

7*x *+ 3 < 5*x *+ 9

Subtract 3 to the both side of the equation

7*x *+ 3 - 3 < 5*x *+ 9 -3

7x < 5x + 6

Subtract 5x on both sides of the equation

7x - 5x < 5x + 6 - 5x

2x<6

Divide by 2 on both sides of the equation

2x / 2 < 6 / 2

x < 3

So, the solution is (-infinity, 3).

**Example problem 3:**

Solve the linear inequality: 5*x *– 3 < 3*x *+1

**Solution:**

5*x *– 3 < 3*x *+1

Add 3 to the both side of the equation

5*x *– 3+3 < 3*x *+1+3

5x < 3x + 4

Subtract 3x on both sides of the equation

5x - 3x < 4

2x < 4

Divide by 2 on both side of the equation

2x / 2 < 4 / 2

x<2

So, the Inequality range is (-infinity, 2)

**Example problem 4:**

Solve the linear inequality: 37 – (3*x *+ 5) > 9*x *– 8 (*x *– 3)

**Solution:**

37 – (3*x *+ 5) > 9*x *– 8 (*x *– 3)

37- 3x - 5 > 9x - 8x + 24

-3x + 32 > 1x + 24

Subtract x on both sides of the equation

-3x+32- x > 1x + 24 - x

-4x +32 > 24

Subtract 32 on both sides of the equation

-4x + 32-32 > 24 - 32

-4x > -8

Divide by -4 on both sides of the equation and change the inequality sign

-4x / -4 < -8 / -4

x <2

So, the solution is (-infinity, 2)

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