College Algebra: Section 1.5a (Packet #9)
By the end of the section, you should be able to:
- Classify the types of equations as conditional, contradiction, or identity.
- Solve a linear equation in one variable
- Solve absolute value equations
Definitions:
- Identity: an equation that is always true for any allowable variable (ex. x = x)
- Contradiction: an equation that is never true for any allowable variable (ex. y = y + 1)
- Conditional: an equation that is sometimes true for any allowable variable (ex. x = 4 is true when x is 4)
- Solution: any value that can replace the variable and make a true statement.
- Solution set: the set of all solutions to an equation.
- Linear equation in one variable/First-degree equations: any equation of the form ax + by = 0
- a and b must be real numbers and a ≠ 0
- Absolute Value: (ex. |x|) the distance a real number x is away from zero.
- Distance is always positive
- Extraneous Solution: a solution found from an original equation that is not a solution to the original equation.
Solving Linear Equations in One Variable
🎥Example 1 🎥Examples 2&3
🎥Example 1 🎥Examples 2&3
Absolute Value Notation
Things to remember about solving Absolute Value Equations:
- When solving absolute value equations ALWAYS get the absolute value by itself on one side of the equation FIRST.
- Once the absolute value is isolated create two equations
- The absolute value set equal to the rest of the equation
- The absolute value set equal to the OPPOSITE of the rest of the equation (multiply a negative through)
- DO NOT move anything outside of the absolute value bars unless you have split the absolute value into two equations.
- When moving parts from one side of an equation to the other work furthest away from the absolute value first. In example 1 we moved the 3 (addition/subtraction) first, then we moved the 2 (multiplication/division).
- Remember to check your solution(s) in the original equation (the equation given in the problem)
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