Solving Inequalities In One Variable

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

1. 2x+5 < 3x+4
2. 2(7x-2) > 9x+6

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

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