Solving Linear Equations and Inequalities

An equality is a mathematical statement which states that two expressions are not equal. Solving a system of equations or inequalities means finding the set of all ordered pairs of numbers that makes each equation or inequality true. A linear inequality contains linear expressions on one or both sides of the inequality. The inequality is expressed by four different symbols >, < , ≥,  ≤ . An inequality is the same as an equation except it can have a less than sign, a greater than sign, a less than or equal to sign, or a greater than equal to sign. The solution set of an inequality is the set of numbers that make the inequality true.

Properties of Inequalities

The following properties are used in solving inequalities.
1. If a,b and c are real numbers and if a > b, then a + c > b + c.
2. If a,b and c are real numbers and if a > b, then a - c > b - c.
3. For real numbers a, b, c and c > 0, if a > b, then ac > bc.
4. For real numbers a, b, c and c < 0, if a > b, then ac < bc.
5. For real numbers a, b, c and c > 0, if a > b, then $\frac{a}{c} > \frac{b}{c}$.
6. For real numbers a, b, c and c < 0, if a > b, then $\frac{a}{c} < \frac{b}{c}$.

Solving Linear Equations and Inequalities with one variable

A linear equation in one variable x is an equation that can be written in the form ax + b = 0, where a and b are real numbers with a$\neq$0. To solve a linear equation, isolate x on the one side of the equation by creating a sequence of equivalent equations. Solving a linear inequality in one variable is much like solving a linear equation in one variable. To solve the linear inequality, firstly isolate the variable on one side using transformations that produce equivalent inequalities. When each side of inequality is multiply or divide by a negative number then inequality sign get reversed. The steps used in solving an inequality are similar to the those used for solving equations. We need to get an equivalent inequality with the variable x isolated.

Solved Examples

Question 1: Solve, 10x - 5 = 15

Solution:
Step 1:

10x - 5 = 15

=> 10x - 5 + 5 = 15 + 5

Step 2:
Divide each side by 10

=> 10x = 20

=> x = 2

Question 2: Solve the inequality 5x - 2 ≤ 3
Solution:
Step 1:

5x - 2 $\leq$ 3

Add 2 to each side to eliminate -2 from the left side.

=> 5x - 2 + 2 $\leq$ 3 + 2

=> 5x $\leq$ 5

Step 2:

Divide each side by 5

=> $\frac{5x}{5}\leq \frac{5}{5}$

=> x $\leq$ 1

The solution can be given in interval form and also shown on the number line.

=>The solution as an interval is (-$\infty$,1] .

Step 3:

The solution on the number line is: Question 3: Solve the inequality 4a + 10 < 6a + 11

Solution:
Step 1:

First let us remove 10 on the left side

Subtract 10 from each side

4a + 10 - 10 < 6a + 12 - 10

=> 4a < 6a + 2

Step 2:

Subtract 6a from each side

=> 4a - 6a < 6a - 6a + 2

=> - 2a < 2

Dividing the inequality by -2, turns the inequality around as

=> a > -1

The solution as an interval is (-1, ∞)

Step 3:

The solution is shown on number line as follows: Question 4: Solve -4 + 10x = 36

Solution:
Step 1:

-4 + 10x = 36

=> $\frac{10x}{10} = \frac{40}{10}$