1. Linear Equations and Absolute Value Equations
Chapter 8
A Transition
Section 1
Linear Equations and Absolute Value Equations

2. Solving Linear Equations
Simplify each side of the equation completely.
Use the distributive property to clear any parentheses.
If there are fractions in the equation, multiply both sides of the equation by the LCM of the denominators to clear the fractions from the equation.
Combine any like terms that are on the same side of the equation.
Collect all variable terms on one side of the equation.
Collect all constant terms on the other side of the equation.
Divide both sides of the equation by the coefficient of the variable term.
Check your solution.

3. Solving Linear Equations
Solve

4. Solving Linear Equations
Solve

5. Solving Linear Equations
Solve

6. Solving Linear Equations
Solve
LCM: 12

7. Contradictions
A contradiction is an equation that has no solution, so its solution set is the empty set { }.
The empty set is also known as the null set, and is denoted by the symbol

8. Contradictions
Solve

9. Identities
An identity is an equation that is always true. The solution set for an identity is the set of all real numbers, denoted . An identity has infinitely many solutions.

10. Identities
Solve

11. Solving Absolute Value Equations
For any expression X and any positive number a, the solutions to the equation
can be found by solving the two equations and

12. Solve |x| = 6.
The absolute value of a number is a measure of the distance between 0 and that number on a real number line.
We are looking for a number that is 6 units away from zero.
The solution set is {–6, 6}.

13. Solving Absolute Value Equations
Solve: or

14. Solving Absolute Value Equations
Solve:
An absolute value is nonnegative, so the equation has no solution:

15. Solving Absolute Value Equations Involving Two Absolute Values
For any expressions X and Y, the solutions to the equation can be found by solving the two equations
and

16. Solving Absolute Value Equations Involving Two Absolute Values
Solve: or