Download 9.6 Linear Inequalities and Problem Solving

9.6 Linear Inequalities and Problem Solving
A linear inequality in one variable is an equation that can be written in the form ax + b < c where a, b, and
c are real numbers, a ≠ 0. Note: the < symbol could be replaced by > or ≤ or ≥.
Graphing solutions to linear inequalities in one variable
•
•
•
Use a number line
Use a closed circle at the endpoint of an interval if you want to include the point
Use an open circle at the endpoint if you DO NOT want to include the point
7
-4
Represents the set {x½x £ 7}
Represents the set {x½x > – 4}
Addition Property of Inequalities
If a, b, and c are real numbers, then
<
and
+ <
+
are equivalent inequalities
Multiplication Property of Inequalities
1. If a, b, and c are real numbers, and c is positive, then
<
and
<
are equivalent inequalities
2. If a, b, and c are real numbers, and c is negative, then
<
and
>
are equivalent inequalities
To Solve Linear Inequalities in One Variable
Step 1:
If an inequality contains fractions, multiply both sides by the LCD to
clear the inequality of fractions.
Step 2:
Use distributive property to remove parentheses if they appear.
Step 3:
Simplify each side of inequality by combining like terms.
Step 4:
Get all variable terms on one side and all numbers on the other side by
using the addition property of inequality.
Step 5:
Get the variable alone by using the multiplication property of inequality.
Examples: Solve the inequality and graph the solution set.
> −4(5 − ) − 12
1. 3x + 9 ≥ 5(x − 1)
2. 7( − 2) +
3. −4 + 7 > −9
4. −3 ≤ −4 + 5
Solve the compound inequality. Write the answer using algebraic and interval notation. Graph the
solution set.
5. −6 < 2 + 8 ≤ 4
6. −14 < 2 ≤ −4
7. −15 < −3 + 8 ≤ 17
8. −6 < −3 ≤ 21