Week 28: Rational equations and absolute value inequalities

Field Book

101 to 104

Due

Test 25 due

Method

Paper first, every step

Two things this week that both hinge on checking your work. Rational equations can produce answers that break the original problem by making a denominator zero, and those answers have to be thrown out no matter how correct the algebra was. Absolute value inequalities split into two cases, and which two depends on whether the sign is less than or greater than.

What you have to be able to do

Factor a denominator to build the LCD
Solve a rational equation and reject any excluded value
Solve an absolute value inequality

The procedures

Rational equations with factorable denominators

Factor every denominator first. The LCD is built from those factors.
Multiply every term by the LCD to clear the fractions, then solve normally.
Check each answer against the original denominators. Any value that makes one zero is not a solution, no matter how it got there.

Absolute value inequalities

Less than makes a sandwich: the expression sits between the negative and the positive of the number, joined by and.
Greater than splits apart: two separate statements joined by or, one going each direction.
Remember it as: less than is and, greater than is or.

Worked

WORKED

Solve the absolute value of x is less than 5, then greater than 3.
Less than 5 becomes a sandwich between −5 and 5.
Greater than 3 splits into two directions joined by or.
ANSWER: −5 < x < 5 and x < −3 or x > 3

WORKED

Solve 1x + 12 = 34.
The LCD is 4x. Multiply every term: 4 + 2x = 3x.
So x = 4. Check: it does not make any denominator zero.
ANSWER: x = 4

Terms

Term	Meaning
excluded value	a value that makes a denominator zero and cannot be a solution
LCD	the least common denominator, built from the factors of every denominator
absolute value	distance from zero, always positive

Check yourself

1. Solve the absolute value of x is less than 2

2. Solve the absolute value of x is greater than 6

3. Why is x = 0 rejected in an equation containing 1x?

reveal answers

−2 < x < 2 · x < −6 or x > 6 · it makes the denominator zero

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.