# the values of the variable for which the given expression is defined as real number. ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.7, Problem 102E
To determine

## To find: the values of the variable for which the given expression is defined as real number.

Expert Solution

x(2,1] .

### Explanation of Solution

Given:

The given expression is 1x2+x4 .

Concept used:

Guidelines for solving nonlinear inequality:

1. Move all terms to one side.
2. Factor the non-zero side of the inequality.
3. Find the value for which each factor is zero. The number will divide the real lines into interval. List the interval determined by these numbers.
4. Make a table or diagram by using test values of the signs of each factor on each interval. In the last row of the table determining the sign of the product of these factors.
5. Determine the solution of the inequality from the last row of the sign table.

Calculation:

Since the given expression 1x2+x4 has a fourth root expression, so the radicand must be non-negative.

The domain of x25x14 is greater than or equal to 0.

Thus, the factor is 1x2+x0

First to find the zeros of the expression in the numerator and demniminator, then

(1x)=0x=1(x+2)=0x=2

From the three zeros above, it extracts the following intervals:

(,2)(2,1)(1,)

Now, make a table by using test values of the signs of each factor on each interval.

 Interval (−∞,−2) (−2,1) (1,∞) (x−7) − − + (x+2) − + + (x+2)(x−7) + − +

As it is seen that the solution set is greater than or equal to 0.

Hence, the solution set is x(2,1] .

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