2. Solving Linear Equations and Inequalities
2.5 Solve Inequalities
Lynn Marecek and MaryAnne Anthony-Smith
Learning Objectives
By the end of this section, you will be able to:
- Graph inequalities on the number line
- Solve inequalities using the Subtraction and Addition Properties of inequality
- Solve inequalities using the Division and Multiplication Properties of inequality
- Solve inequalities that require simplification
- Translate to an inequality and solve
Graph Inequalities on the Number Line
Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
What about the solution of an inequality? What number would make the inequality > true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality > .
We show the solutions to the inequality > on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of > is shown in (Figure). Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.
The graph of the inequality is very much like the graph of > , but now we need to show that 3 is a solution, too. We do that by putting a bracket at , as shown in (Figure).
Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.
EXAMPLE 1
Graph on the number line:
a) b) < c) >
Solution
a)
This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at to show that it is included.
b) <
This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at to show it is not included.
c) >
This means all numbers greater than , but not including . We shade in all the numbers on the number line to the right of , then put a parenthesis at to show it is not included.
TRY IT 1
Graph on the number line: a) b) > c) <
Show answer
a)
b)
c)
We can also represent inequalities using interval notation. As we saw above, the inequality > means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express > as The symbol is read as ‘infinity’. It is not an actual number. (Figure) shows both the number line and the interval notation.
The inequality means all numbers less than or equal to 1. There is no lower end to those numbers. We write in interval notation as . The symbol is read as ‘negative infinity’. (Figure) shows both the number line and interval notation.
Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in (Figure).
EXAMPLE 2
Graph on the number line and write in interval notation.
a) b) < c)
a)
| Shade to the right of , and put a bracket at . | |
| Write in interval notation. |
b)
| Shade to the left of , and put a parenthesis at . | |
| Write in interval notation. |
c)
| Shade to the left of , and put a bracket at . | |
| Write in interval notation. |
TRY IT 2
Graph on the number line and write in interval notation:
a) > b) c)
Show answer
a)
b)
c)
Solve Inequalities using the Subtraction and Addition Properties of Inequality
The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.
Properties of Equality
Similar properties hold true for inequalities.
| For example, we know that −4 is less than 2. | |
| If we subtract 5 from both quantities, is the left side still less than the right side? |
|
| We get −9 on the left and −3 on the right. | |
| And we know −9 is less than −3. | |
| The inequality sign stayed the same. |
Similarly we could show that the inequality also stays the same for addition.
This leads us to the Subtraction and Addition Properties of Inequality.
Properties of Inequality
We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality > , the steps would look like this:
| > | |
| Subtract 5 from both sides to isolate . | > |
| Simplify. | > |
Any number greater than 4 is a solution to this inequality.
EXAMPLE 3
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
| Add to both sides of the inequality. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. |
TRY IT 3
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Show answer
The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).
Properties of Equality
Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?
Consider some numerical examples.
| Divide both sides by 5. | Multiply both sides by 5. | ||
| Simplify. | |||
| Fill in the inequality signs. |
Does the inequality stay the same when we divide or multiply by a negative number?
| Divide both sides by −5. | Multiply both sides by −5. | ||
| Simplify. | |||
| Fill in the inequality signs. |
When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.
Here are the Division and Multiplication Properties of Inequality for easy reference.
Division and Multiplication Properties of Inequality
When we divide or multiply an inequality by a:
- positive number, the inequality stays the same.
- negative number, the inequality reverses.
EXAMPLE 4
Solve the inequality < , graph the solution on the number line, and write the solution in interval notation.
| Divide both sides of the inequality by 7. Since > , the inequality stays the same. |
|
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. |
TRY IT 4
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Show answer
>
EXAMPLE 5
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Solution
| Divide both sides of the inequality by −10. Since < , the inequality reverses. |
|
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. |
TRY IT 5
Solve each inequality, graph the solution on the number line, and write the solution in interval notation.
<
Show answer
>
Solving Inequalities
Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.
Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”
EXAMPLE 6
Solve the inequality < , graph the solution on the number line, and write the solution in interval notation.
| Multiply both sides of the inequality by . Since > , the inequality stays the same. |
|
| Simplify. | |
| Rewrite the variable on the left. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. |
TRY IT 6
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Show answer
EXAMPLE 7
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
| Multiply both sides of the inequality by . Since < , the inequality reverses. |
|
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. |
TRY IT 7
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Show answer
Solve Inequalities That Require Simplification
Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.
EXAMPLE 8
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
| Subtract from both sides to collect the variables on the left. | |
| Simplify. | |
| Divide both sides of the inequality by −5, and reverse the inequality. | |
| Simplify. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. |
TRY IT 8
Solve the inequality , graph the solution on the number line, and write the solution in interval notation.
Show answer
EXAMPLE 9
Solve the inequality > , graph the solution on the number line, and write the solution in interval notation.
| Simplify each side as much as possible. | > |
| Distribute. | > |
| Combine like terms. | > |
| Subtract from both sides to collect the variables on the left. | > |
| Simplify. | > |
| Add 36 to both sides to collect the constants on the right. | > |
| Simplify. | > |
| Divide both sides of the inequality by 4; the inequality stays the same. | > |
| Simplify. | > |
| Graph the solution on the number line. | |
| Write the solution in interal notation. |
TRY IT 9
Solve the inequality > , graph the solution on the number line, and write the solution in interval notation.
Show answer
Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.
EXAMPLE 10
Solve the inequality < , graph the solution on the number line, and write the solution in interval notation.
| Simplify each side as much as possible. | < |
| Distribute. | < |
| Combine like terms. | < |
| Subtract from both sides to collect the variables on the left. | < |
| Simplify. | < |
| The ’s are gone, and we have a true statement. | The inequality is an identity. The solution is all real numbers. |
| Graph the solution on the number line. | |
| Write the solution in interval notation. |
TRY IT 10
Solve the inequality > , graph the solution on the number line, and write the solution in interval notation.
Show answer
EXAMPLE 11
Solve the inequality > , graph the solution on the number line, and write the solution in interval notation.
| Multiply both sides by the LCD, 24, to clear the fractions. | |
| Simplify. | |
| Combine like terms. | |
| Subtract from both sides to collect the variables on the left. | |
| Simplify. | |
| The statement is false! | The inequality is a contradiction. |
| There is no solution. | |
| Graph the solution on the number line. | |
| Write the solution in interval notation. | There is no solution. |
TRY IT 11
Solve the inequality > , graph the solution on the number line, and write the solution in interval notation.
Show answer
Translate to an Inequality and Solve
To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.
Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.
(Figure) shows some common phrases that indicate inequalities.
| > | < | ||
|---|---|---|---|
| is greater than | is greater than or equal to | is less than | is less than or equal to |
| is more than | is at least | is smaller than | is at most |
| is larger than | is no less than | has fewer than | is no more than |
| exceeds | is the minimum | is lower than | is the maximum |
EXAMPLE 12
Translate and solve. Then write the solution in interval notation and graph on the number line.
Twelve times c is no more than 96.
| Translate. | |
| Solve—divide both sides by 12. | |
| Simplify. | |
| Write in interval notation. | |
| Graph on the number line. |
TRY IT 12
Translate and solve. Then write the solution in interval notation and graph on the number line.
Twenty times y is at most 100
Show answer
EXAMPLE 13
Translate and solve. Then write the solution in interval notation and graph on the number line.
Thirty less than x is at least 45.
| Translate. | |
| Solve—add 30 to both sides. | |
| Simplify. | |
| Write in interval notation. | |
| Graph on the number line. |
TRY IT 13
Translate and solve. Then write the solution in interval notation and graph on the number line.
Nineteen less than p is no less than 47
Show answer
Key Concepts
- Subtraction Property of Inequality
For any numbers a, b, and c,
if < then < and
if > then > - Addition Property of Inequality
For any numbers a, b, and c,
if < then < and
if > then > - Division and Multiplication Properties of Inequality
For any numbers a, b, and c,
if < and > , then < and > .
if > and > , then > and > .
if < and < , then > and > .
if > and < , then < and < .
- When we divide or multiply an inequality by a:
- positive number, the inequality stays the same.
- negative number, the inequality reverses.
2.5 Exercise Set
In the following exercises, graph each inequality on the number line.
-
- >
- <
-
- >
In the following exercises, graph each inequality on the number line and write in interval notation.
-
- >
-
- <
- >
- <
- <
- >
- <
- >
- <
- >
- <
- >
- <
- >
- Ninety times c is less than 450.
- Ten times y is at most .
- Six more than k exceeds 25.
- Twelve less than x is no less than 21.
- Negative two times s is lower than 56.
- Fifteen less than a is at least .
- The maximum height, h, of a fighter pilot is 77 inches. Write this as an inequality.
Answers
| 1.
a. b. c. |
2.
a. b. c. |
3.
a. b. c. |
| 4.
a. b. c. |
5. | 6. |
| 7. | 8. | 9. |
| 10. | 11. | 12. |
| 13. | 14. | 15. |
| 16. | 17. | 18. |
| 19. | 20. | 21. |
| 22. | 23. | 24. |
| 25. | 26. | 27. |
| 28. | 29. | 30. |
| 31. | 32. | 33. |
| 34. | 35. | 36. |
| 37. | 38. |