Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 7 - Review Exercises - Page 878: 51


The graph is shown below: (no solution)

Work Step by Step

Let us consider the inequalities $\begin{align} & 2x+y<4 \\ & 2x+y>6 \\ \end{align}$ Put the equals symbol in place of the inequality and rewrite the equation as follows: $\begin{align} & 2x+y=4 \\ & 2x+y=6 \\ \end{align}$ Now, take origin $\left( 0,0 \right)$ as a test point for the equations and check the region in the graph to shade: $\begin{align} & 2x+y<4\text{ and }2x+y>6 \\ & 2\left( 0 \right)+0<4\text{ and }2\left( 0 \right)+0>6 \\ & 0<4\text{ and }0>6 \\ \end{align}$ We see that the test point satisfies the first inequality but it does not satisfy the second inequality, so the shaded region will not contain the origin. Thus, the shaded region for both equations is in the opposite direction and there is no region in common. Hence, the system of inequalities has no solution. Thus, the graph of the given inequality is plotted and there is no solution for the provided system of inequalities.
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