PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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Chapter P.7, Problem 37E
Solution

a.

The possible width in inches of the given rectangles if their perimeters are less than 200 in.

  0 in.<x<34 in.

Given:

The rectangles have length 2 in. less than twice their width.

Concept Used:

  • Perimeter of a rectangle is, P=2Length+Width .
  • If there is the strict inequality symbol < or> , then use a parenthesis on the interval
  • If there is the inequality symbol ≤ or≥ , then use a bracket on the interval
  • To graph the endpoint with strict inequality < or>, use the open parenthesis or a hollow circle at that endpoint.
  • To graph the endpoint with inequality symbol ≤ or≥, use the bracket or a solid circle at that endpoint.
  • Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

Let x be the width of the rectangle.

Since the length is 2 in. less than twice the width, the length of the rectangle is 2x−2 .

In order find the possible width, solve the inequality 2x+2x−2<200 as shown below

  2x+2x−2<2002x+2x−22<2002x+2x−2<1003x−2<1003x<102x<1023x<34

Since the width of the rectangle must be greater than 0, the possible width of the given rectangles is 0 in.<x<34 in. .

b.

The possible width in inches of the given rectangles if their areas are less than or equal to 1200 square inches.

  0 in.<x≤25 in.

Given:

The rectangles have length 2 in. less than twice their width.

Concept Used:

  • Area of a rectangle is, A=Length×Width .
  • If there is the strict inequality symbol < or> , then use a parenthesis on the interval
  • If there is the inequality symbol ≤ or≥ , then use a bracket on the interval
  • To graph the endpoint with strict inequality < or>, use the open parenthesis or a hollow circle at that endpoint.
  • To graph the endpoint with inequality symbol ≤ or≥, use the bracket or a solid circle at that endpoint.
  • Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

Let x be the width of the rectangle.

Since the length is 2 in. less than twice the width, the length of the rectangle is 2x−2 .

In order find the possible width, solve the inequality x⋅2x−2≤1200 as shown below

  x⋅2x−2≤12002x2−2x−1200≤02x2−x−600≤0x2−x−600≤0x+24x−25≤0⇒x∈−24,25

Since the width of the rectangle must be greater than 0, the possible width of the given rectangles is 0 in.<x≤25 in. .

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Chapter P.7, Problem 36E
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Chapter P.7, Problem 38E

Chapter P Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

Ch. P.1 - Prob. 1QRCh. P.1 - Prob. 2QRCh. P.1 - Prob. 3QRCh. P.1 - Prob. 4QRCh. P.1 - Prob. 5QRCh. P.1 - Prob. 6QRCh. P.1 - Prob. 7QRCh. P.1 - Prob. 8QRCh. P.1 - Prob. 9QRCh. P.1 - Prob. 10QR
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