# Linear & Absolute Value Equations

## Overview

• Section 1.1 in the textbook:

– Solving Linear Equations

– Contradictions and Identities

– Absolute Value Equations

## Solving Linear Equations

## Definition of a Linear Equation

**• Linear Equation:** where all the variables

are raised to the first power

**• Linear Equation (in one variable):** has

the form ax + b = 0 where a and b are

constants and a ≠ 0

## Solving Linear Equations in One Variable

• To solve more complex equations :

– Apply the distributive property

– Combine like terms

– Isolate the variable on one side of the equation

– Apply the Addition Property of Equality

– Apply the Multiplication Property of Equality

– Don’t forget to simplify

– Check

## Solving Linear Equations in One

Variable (Example)

**Ex 1:** Solve: 3(x – 5) – (x + 2) = 4

**Ex 2:** Solve: -2(7 – 3x) + 2x = 2 – (1 – x)

## Solving Linear Equations with

Fractions

• While it is certainly possible to work with

fractions in an equation , it is often easier to

eliminate them

• How do we add two ** UNLIKE ** fractions?

– Look for an LCD

• We can use the LCD to help us eliminate the

fractions in an equation

• What kind of mathematical statement is adding

½ + ¼?

– Expression

• The rules are a bit different with an

equation

• What must always be remembered when

performing operations on equations?

– “What you do to one side, you must do to the

other”

• How can we use the LCD to help us solve

an equation with fractions?

## Solving Linear Equations with

Fractions (Example)

**Ex 3: **Solve:

**Ex 4:** Solve

**Contradictions & Identities**

## Identities and No Solutions

• Can only happen when the variable drops

out on **BOTH** sides of the equation

• Determine whether the resulting statement

is true or not:

– If yes, then the equation has an infinite

number of solutions and we say the solution is

**all real numbers**

• Also called an identity

– If no, then the equation has **no solution**

• Also called a contradiction

## Identities and No Solutions

(Example)

**Ex 5:** Solve: 2(5x – 2) – 2 = 20x – 1 – 5(2x + 1)

**Ex 6:** Solve:

**Absolute Value Equations**

• Absolute Value: Distance from 0 as

viewed on a number line

– Thus can only be positive or 0

• If |x| = a (a > 0), then x = a or x = -a

-> {a, -a} (written in set notation)

• If |x| = 0, then x = 0 -> {0}

• If |x| = a (a < 0), what is the solution? Ø

• Before applying the definition of absolute

value, the absolute value must be

**ISOLATED** first

–** VERY** important!

– The absolute value must be **ISOLATED**

before the definition of absolute value can be

applied!

## Absolute Value Equations

(Example)

**Ex 7:** Solve:

**Ex 8:** Solve:

**Ex 9:** Solve:

**Ex 10:** Solve:

## Summary

• After studying these slides, you should

know how to do the following:

– Solve Linear Equations in One Variable

including Identities and Contradictions

– Solve Absolute Value Equations

• Additional Practice

– See the list of suggested problems for 1.1

• Next lesson

– Formulas and Applications (Section 1.2)

Prev | Next |