Beginning Algebra, 11th Edition

Solving an Equation That Has Infinitely Many Solutions

Solve

Distributive property

Subtract 5x.

Combine like terms.

Add 15.

True

Solution set: {all real numbers}

Since the last statement is true, anyreal number is a solution. We could have

predicted this from the second line in the solution,

This is true for anyvalue of x.

Try several values for xin the original equation to see that they all satisfy it.

An equation with both sides exactly the same, like is called an identity.

An identity is true for all replacements of the variables. As shown above, we write the

solution set as {all real numbers}.

0 = 0,

5 x- 15 = 5 x- 15.

10 = 02

0 = 0

- 15 + 15 =- 15 + 15

- 15 =- 15

5 x- 15 - 5 x= 5 x- 15 - 5 x

5 x- 15 = 5 x- 15

5 x- 15 = 51 x- 32

5 x- 15 = 51 x- 32.

EXAMPLE 9

SECTION 2.3 More on Solving Linear Equations 103

The table summarizes the solution sets of the equations in this section.

NOW TRY
EXERCISE 9
Solve.

• 31 x- 72 = 2 x- 5 x+ 21

NOW TRY
EXERCISE 10
Solve.

• 4 x+ 12 = 3 - 41 x- 32

Notice that the

variable “disappeared.”

Solving an Equation That Has No Solution

Solve

Distributive property

Combine like terms.

Subtract 5x.

False

There is no solution. Solution set:

A false statement results. The original equation, called a contradiction,has no

solution. Its solution set is the empty set,or null set,symbolized 0.

13 = 42

0

3 = 4

5 x+ 3 - 5 x= 5 x+ 4 - 5 x

5 x+ 3 = 5 x+ 4

2 x+ 3 x+ 3 = 5 x+ 4

2 x+ 31 x+ 12 = 5 x+ 4

2 x+ 31 x+ 12 = 5 x+ 4.

EXAMPLE 10

CAUTION In Example 9,do not write as the solution set. While 0 is a

solution, there are infinitely many other solutions. For to be the solution set, the

last line must include a variable, such as x, and read x 0 , not 0  0.

506

506

Again, the variable

“disappeared.”

CAUTION DO NOTwrite 506 to represent the empty set.

Type of Equation Final Equation in Solution Number of Solutions Solution Set

Conditional a number One a number

(See Examples 1–8.)

Identity A true statement with no Infinite all real numbers

(See Example 9.) variable, such as

Contradiction A false statement with no None

(See Example 10.) variable, such as 3= 4

0

0 = 0

5 6

x= 5 6

NOW TRY ANSWERS

9. 5 all real numbers 6 10. 0

NOW TRY

NOW TRY