Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 7 - Algebra: Graphs, Functions, and Linear Systems - Chapter Summary, Review, and Test - Review Exercises - Page 482: 56

Answer

See below:

Work Step by Step

Graph each inequality on the same plane. Step 1: Convert the inequality to an equation by replacing inequality by = sign. Equation 1: \[3x-y\le 6\] \[3x-y=6\] Equation 2: \[x+y\ge 2\] \[x+y=2\] Step 2: Graph the equation using intercept form or slope-intercept form. Equation 1: \[3x-y\le 6\] Make \[x=0\] to find the y-intercept, \[\begin{align} & 3x-y=6 \\ & 0-y=-6 \end{align}\] Make \[y=0\] to find the x-intercept, \[\begin{align} & 3x-y=6 \\ & 3x-0=6 \\ & x=2 \end{align}\] The line passes through \[\left( 0,-6 \right)\] and\[\left( 2,0 \right)\]. The line is solid since the inequality contains the ≤symbol. Equation 2: \[x+y\ge 2\] Make \[x=0\] to find the y-intercept, \[\begin{align} & x+y=2 \\ & 0+y=2 \\ & y=2 \end{align}\] Make \[y=0\] to find the x-intercept, \[\begin{align} & x+y=2 \\ & x+0=2 \\ & x=2 \end{align}\] The line passes through \[\left( 0,2 \right)\] and\[\left( 2,0 \right)\]. The line is solid since the inequality contains the ≥symbol. Step 3: Choose a test point. The origin \[\left( 0,0 \right)\]be the test point. Substituting in the inequality Equation 1: \[3x-y\le 6\] \[0\le 6\] The statement is true. The graph contains the test point. Equation 2: \[x+y\ge 2\] \[0\ge 2\] The statement is false. The graph does not contain the test point.
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