Download Unit 1 Assignment 10 2.9 Solving Linear Inequalities

Unit 1 Assignment 10
2.9 Solving Linear Inequalities
An inequality is a statement that contains of the symbols: < , >, ≤ or ≥.
Equation
Inequality
2(x – 5) = x + 7
2(x – 5) < x + 7
Solutions of an inequality are any values of the variable that make the inequality true.
For example, some solutions to the inequality x ≥ ‒3 are:
Graphing solutions to linear inequalities in one variable
• Use a number line
• Use a bracket at the endpoint of an interval if you want to include the point
• Use a parenthesis at the endpoint if you DO NOT want to include the point
Represents the set {xx  7}
Represents the set {xx > – 4}
Different ways to write the Solutions to Linear Inequalities:
Example 1:
Inequality:
Example 2:
x≥3
Inequality:
Graph:
Graph:
Interval notation:
Interval notation: (‒∞, 5)
Addition Property of Inequality
If a, b, and c are real numbers, then a < b and a + c < b + c are equivalent inequalities.
Example 3: Solve x – 2 ≤ ‒10
Multiplication Property of Inequality
•
If a, b, and c are real numbers, and c is positive, then
a < b and ac < bc are equivalent inequalities.
•
If a, b, and c are real numbers, and c is negative, then
a < b and ac > bc are equivalent inequalities.
Example 4: Solve
– 6x > 24
Solving Linear Inequalities
1. If there are fractions, multiply both sides of the inequality by the LCD (Least
Common Denominator)
2. If there are any parentheses, remove them by using the distributive property
3. Combine any like terms that are on the same side of the inequality
4. Sort variables to one side and numbers (constants) to the other side of the
inequality by adding or subtracting terms.
5. Divide both sides by the coefficient of the variable. If the coefficient was negative,
you must reverse your inequality sign.
6. Graph your solution and/or write in interval notation if requested.
Example 5: Solve the inequality. Graph the solution set and write in interval notation.
8x – 4 ≤ 10x – 4x
Solve the following inequalities. Graph the solution set and write in interval notation.
6. – 3x + 1 > 2(1 – x)
7. – 3(x – 4) – 3x < – (x – 5) + 1
Compound Inequalities
A compound inequality contains two inequality symbols.
0x–3<8
This means 0  x – 3 and x – 3 < 8.
To solve the compound inequality, perform operations simultaneously to all three parts of the inequality
(left, middle and right).
Example 8: Graph the inequality. Then write the solution set in interval notation.
–1<x≤4
Solve the following inequalities. Graph the solution set and write it in interval notation.
9. 3 ≤ 3x – 9 < 9
10. – 2 < 2(x – 3) < 4