LINEAR EQUATIONS AND INEQUALITIES IN ONE VARIABLE

Linear equations in one variable :

A linear equation in one variable is any equation that can be written in the form 

ax + b  =  0

(where a and b are real numbers and x is a variable)

It is known as standard form of a linear equation in one variable.

To see more examples

Linear inequalities in one variable :

A linear inequality in one variable is an inequality that can be written in the form

ax + b > c

Also, includes forms with <, ≥ and 

(where a, b, and c are real numbers and x is a variable)

Rules for Solving Inequalities

  • When we add, subtract, multiply or divide any non zero number on both sides of the inequality sign, we don't have to change the inequality sign.
  • When we multiply or divide some negative number on both side of the equal sign, we have to reverse the sign.

To see more examples

Solve the linear equations : 

Example 1 :

x - 3  =  2

Solution : 

Add 3 to each side of the equation.

x - 3 + 3  =  2 + 3

x  =  5

Example 2 : 

3x - 2(x - 1)  =  5

Solution : 

Simplify the left side of the equation.

3x - 2(x - 1)  =  5

3x - 2x + 2  =  5

Combine the like terms.

x + 2  =  5

Subtract 2 from each side of the equation.

x + 2 - 2  =  5 - 2

x  =  3

Example 3 : 

5(x - 3) - 7(6 - x)  =  24 - 3(8 - x) - 3

Solution : 

Simplify each side of the equation. 

5(x - 3) - 7(6 - x)  =  24 - 3(8 - x) - 3

5x - 15 - 42 + 7x  =  24 - 24 + 3x - 3

Combine the like terms. 

12x - 57  =  3x - 3

Subtract 3x from each side of the equation. 

12x - 57 - 3x  =  3x - 3 - 3x

9x - 57  =  - 3

Add 57 to each side of the equation. 

9x - 57 + 57  =  - 3 + 57

9x  =  54

Divide each side by 9. 

9x / 9  =  54 / 9

x  =  6

Solve the linear inequalities :

Example 4 :

2(2x + 3) - 10  ≤  6(x - 2)

Solution :

 2(2x + 3) - 10  ≤  6(x - 2)

4x + 6 - 10  ≤  6x - 12

4x - 4  ≤  6x - 12

Subtract 6x from each side. 

-2x - 4  ≤  - 12

Add 4 to each side.

-2x  ≤  - 8

Divide each side by (-2). 

x  ≥  4

So, the solution set is [4, ∞)

Example 5 :

3x - 7  >  x + 1

Solution :

3x - 7  >  x + 1

Subtract x from each side. 

2x - 7  >  1

Add 7 to each side. 

2x  >  8

Divide each side by 2. 

x  >  4

So, the solution set is (4, ∞)

Example 6 :

- (x - 3) + 4  <  5 - 2x

Solution :

- ( x - 3) + 4  <  5 - 2x

-x + 3 + 4  <  5 - 2x 

-x + 7  <  5 -2x

Add 2x to each side.

x + 7  <  5

Subtract 7 from each side. 

x  <  - 2

So, the solution set is (-∞, -2). 

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