Solving Inequalities with Examples and Numerical

Key Concepts

• Graph the solutions of an inequality.

• Graph to solve an inequality.

• Substitute to solve an inequality.

Introduction: 

Equality vs inequality: 

An equation shows when expressions are equal. Equations use equal signs (=).  

An inequality is a statement that uses the greater-than symbol (>), the less-than symbol (<), the greater-than-or-equal-to symbol (≥), or the less-than-or-equal-to symbol (≤).  

Variables in inequality: 

Variables can be used with inequalities. A variable in an inequality stands for all numbers that make the inequality true. 

For example, in the inequality x < 4, the x stands for all numbers less than 4. So x can be 0, 1, 2 or 3. 

The inequality 12 ≤ y + 5 can have solutions y = 7, 8, and 9, since 7 + 5 = 12, 8 + 5 = 13, and 9 + 5 = 14. 

4.7.1 Graph the solutions of an inequality 

Example 1: 

Graph all the solutions of x < 3. 

Solution: 

Step 1: 

To graph x < 3, draw an open circle at 3 on a number line. 

Step 2: 

Find some solutions and plot them on a number line. 

Step 3: 

Start at the open circle and shade the solutions you found. 

4.7.2 Graph to solve an inequality 

Example 2: 

The temperature in a greenhouse should be 57 degrees or higher. Write an inequality to describe the allowable temperature in the greenhouse. 

Solution: 

Write and graph an inequality. 

The possible temperatures in the greenhouse, t, are greater than or equal to 57 degrees. 

Example 3: 

The maximum weight allowed by law in a freight elevator is 1,500 pounds. Let w represent the weight on the elevator. Write an inequality to describe the allowable weight on the elevator. 

Solution: 

Write and graph an inequality. 

The possible weights in the elevator, w, are less than or equal to 1,500 pounds. 

4.7.3 Substitute to solve an inequality 

Example 4: 

John’s family is planning to go on vacation. They are planning to visit any one place which is beyond 250 miles from their home town. Which place, if any, can be chosen for the family vacation? 

Solution: 

Write an inequality to represent the situation. 

Exercise:

1. Name 3 solutions for z> S.

2. Name 3 solutions for x 4.

3. How many solutions does the inequality x > 18 have? Explain.

4. Write the inequality the following graph represents.

5. Write the inequality the following graph represents.

6. Substitute each given value of the variable to find which, if any, is a solution of the inequality.

x < 8 x = 4.2, 5.2, 8.2, 9

7. Substitute each given value of the variable to find which, if any, is a solution of the inequality.

s > 25 s = 24,25, 25.1, 27

8. Substitute each given value of the variable to find which, if any, is a solution of the inequality.

t < 4 t = 0, 4, 5,6

 9. Name three solutions of the inequality t 24.

10. Tina started a graph to show the inequality y < 3.4. Finish labeling the number line and draw the graph.

What have we learned:

• Graph the solutions of an inequality on a number line.

• Graph to solve an inequality on a number line.

• Substitute a set of values in a variable to solve an inequality.

Concept Map