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 brac