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
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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