1. Section 2.3
Solving Linear Equations Involving Fractions and Decimals; Classifying Equations

2. Section 2.3 Objectives
1 Use the Least Common Denominator to Solve a Linear Equation Containing Fractions
2 Solve a Linear Equation Containing Decimals
3 Classify a Linear Equation as an Identity, Conditional, or a Contradiction
4 Use Linear Equations to Solve Problems

3. Solving Equations Containing Fractions
Example:
Solve the equation
Fractions can be removed by multiplying both sides of the equation by the LCD of all the fractions.
The LCD is 10.
Remove all parentheses.
Divide out common factors.
Simplify.
Simplify.
Continued.

4. Solving Equations Containing Fractions
Example continued:
Add –4 to both sides of the equation.
Divide both sides by 10
Check your answer in the original equation. 

5. Solving Equations Containing Decimals
Example:
Solve the equation
Decimals can be eliminated by multiplying both sides of the equation by the LCD of all the decimals.
Distribute before eliminating the decimals.
Combine like terms.
Subtract 3.6x from both sides.
Subtract 5.4 from both sides.
Continued.

6. Solving Equations Containing Decimals
Example continued:
Multiply both sides of the equation by the LCD 10 to eliminate the decimal.
Simplify.
Divide both sides by – 38.
Check your answer in the original equation. 

7. Conditional Equations & Contradictions
A conditional equation is an equation that is true for some values of the variable and false for other values of the variable.
A contradiction is an equation that is false for every replacement value of the variable.
Example:
Solve the equation 4x – 8 = 4x – 1.
4x – 8 = 4x – 1
–8 = –1
Subtract 4x from both sides.
This is a false statement so the equation is a contradiction.
The solution set is the empty set, written { } or .

8. Identities
An identity is an equation that is satisfied for all values of the variable for which both sides of the equation are defined.
Example:
Solve the equation 5x + 3 = 2x + 3(x + 1).
5x + 3 = 2x + 3(x + 1).
5x + 3 = 2x + 3x + 3
Distribute to remove parentheses.
5x + 3 = 5x + 3
Combine like terms.
3 = 3
Subtract 5x from both sides.
This is a true statement for all real numbers x.
The solution set is all real numbers.

9. Solving Problems
You have $5000 to invest, and your financial advisor advises you to put part of the money in a Certificate of Deposit (CD) that earns 2% simple interest compounded annually, and the rest in bonds that earn 4% simple interest compounded annually. To determine the amount x you should invest in the CD to earn $170 interest at the end of one year, solve the equation 0.02x (5000 – x) = 170.
0.02x (5000 – x) = 170
0.02x – 0.04x = 170
Distributive property.
–0.02x = 170
Combine like terms.
–0.02x – 200 = 170 – 200
Isolate x.
–0.02x = –30
Isolate x.

10. Solving Problems –0.02x = –30 2x = –3000 Multiply by 100 x = 1500
Divide by –2
Because x = 1500, you must invest $1500 in Certificate of Deposit to earn $170 in interest at the end of the year.