PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
Expert Solution & Answer
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Chapter P, Problem 63RE
Solution

The solution of the given inequality.

  −∞,−2

Given information:

The inequality given is:

  3x+4≥2 .

Formula used:

  • To solve an inequality, follow the steps below:
    1. Simplify the both sides of inequality.
    2. Take all the terms of variable on one side of the inequality and rest on other side of the inequality.
    3. Make the coefficient of variable 1.
  • For any x :

    •   x=x if x is positive−x if x is negative

  • Properties of inequality:

    1. b≤c⇒b±a≤c±a
    2. b≤c⇒ab≤ac, if a>0
    3. b≤c⇒ab≥ac, if a<0

Calculation:

In order to solve the given inequality:

  3x+4≥2

First simplify the left side using for any x :

  x=x if x is positive−x if x is negative

Now, 3x+4 is positive if

  3x+4≥03x≥−4x≥−43

And negative for

  3x+4≤03x≤−4x≤−43

Thus,

  3x+4=3x+4 if x≥−43−3x−4 if x≤−43

Thus, the inequality can be rewritten as

  for x≥−433x+4≥2

Solve it by taking variable one side and rest other side and simplify, as

  3x+4≥23x≥2−43x<−2x<−23

And case second

  for x≤−43The inequality is:−3x−4≥2−3x≥2+4−3x≥6

Now using the property of inequality b≤c⇒ab≥ac, if a<0 , by dividing both sides by −3 as:

  −3x−3≤6−3x≤−2

Thus, from case 1, the solution of given inequality is for x≥43 solution is x<−23 but that is contradiction since a number cannot be greater than 43 and smaller than −23 at the same time. For x≤43 solution is x≤−2 this can be represented in interval form, as

  −∞,−2

Thus, solution of given inequality is:

  −∞,−2

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Chapter P, Problem 62RE
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Chapter P, Problem 64RE

Chapter P Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

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