## Key Concepts

- Inequality and graph of an inequality
- Inequality with variable on both sides
- Inequalities with infinitely many solutions
- Inequalities with no solutions
- Problems using inequalities

## Inequality

A statement of an order relationship greater than, greater than or equal to, less than, or less than or equal to between two numbers or algebraic expressions is called an inequality.

**a < b** says that a is less than b

**a > b** says that a is greater than b

**a ≤ b** means that a is less than or equal to b

**a ≥ b** means that a is greater than or equal to b

**Example:** Solve for x when 3x – 1 > 5

**Sol:** 3x – 1 > 5

3x – 1 + 1 > 5 + 1

3x > 6

3x/3 > 6/3

x > 2

### Graph an inequality on number line

We can graph an inequality on a number line.

The numbers that are not included are represented using an **open circle**.

**Example:** Graph x > 2

Since x > 2, all the numbers greater than 2 are included.

The numbers that are included are represented using a **closed circle**.

**Example:** Graph x ≤ 3

Since x ≤ 3, all the numbers less than or equal to 3 are included.

### Solving inequalities with variables on both sides

To solve an inequality that has variables on both sides, we collect like terms on the same side of the inequality.

**Example: **Solve – 2x – 5 > 3x – 25

**Sol:** – 2x – 5 + 25 > 3x – 25 + 25

– 2x + 20 > 3x

– 2x + 20 + 2x > 3x + 2x

20 > 5x

20/5 > 5x/5

4 > x

**Therefore, x < 4. **

**Graph of the given inequality**:

### Inequality with no solutions

**Example:** Solve for x if – 7x – 9 ≥ 9 – 7x

**Sol:** – 7x – 9 + 9 ≥ 9 – 7x + 9

– 7x ≥ 18 – 7x

– 7x + 7x ≥ 9 – 7x + 7x

0 ≥ 9

Here, the inequality results in a false statement (0 ≥ 9), so, any value of x when substituted in the original inequality will result in a false statement.

**Therefore, the inequality has no solutions.**

### Inequality with infinitely many solutions

**Example 2:** Solve for x if 2x + 12 > 2(x – 4)

Sol: 2x + 12 > 2x – 8

2x + 12 – 2x > 2x – 8 – 2x

12 > – 8

Here, the inequality is a true statement (12 > – 8), so, the statement is true for all values of x.

**Therefore, the inequality has infinitely many solutions.**

## Exercise

Solve each inequality

- 2x+5 < 3x+4
- 2(7x-2) > 9x+6

Solve each inequality and tell whether it has infinitely many or no solutions

- ¾ x + ¾ x – ½x
**≥**-1 - 1⁄4 x + 3-7/8x < -2
- -5(2x+1) < 24
- 4(3-2x)
**≥**-4

### Concept Map

- To solve inequalities, use the properties of inequalities to isolate the variable.

**Example:** Solve for x when 3x – 1 > 5

**Sol:** 3x – 1 > 5

= 3x – 1 + 1 > 5 + 1

= 3x > 6

= 3x/3 > 6/3

= x > 2

## What have we learned

- Define and solve Inequalities
- Solve an inequality with variable on both sides
- Solve inequalities with infinitely many solutions
- Solve inequalities with no solutions
- Solve problems using inequalities

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