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Mathematics LibreTexts

4.7E: Exercises

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    Practice Makes Perfect

    Verify Solutions to an Inequality in Two Variables

    In the following exercises, determine whether each ordered pair is a solution to the given inequality.

    Exercise \(\PageIndex{1}\)

    Determine whether each ordered pair is a solution to the inequality \(y>x−1\):

    1. \((0,1)\)
    2. \((−4,−1)\)
    3. \((4,2)\)
    4. \((3,0)\)
    5. \((−2,−3)\)

    Exercise \(\PageIndex{2}\)

    Determine whether each ordered pair is a solution to the inequality \(y>x−3\):

    1. \((0,0)\)
    2. \((2,1)\)
    3. \((−1,−5)\)
    4. \((−6,−3)\)
    5. \((1,0)\)
    Answer
    1. yes 
    2. no 
    3. no 
    4. yes 
    5. no

    Exercise \(\PageIndex{3}\)

    Determine whether each ordered pair is a solution to the inequality \(y<x+2\):

    1. \((0,3)\)
    2. \((−3,−2)\)
    3. \((−2,0)\)
    4. \((0,0)\)
    5. \((−1,4)\)

    Exercise \(\PageIndex{4}\)

    Determine whether each ordered pair is a solution to the inequality \(y<x+5\):

    1. \((−3,0)\)
    2. \((1,6)\)
    3. \((−6,−2)\)
    4. \((0,1)\)
    5. \((5,−4)\)
    Answer
    1. yes 
    2. no
    3. no 
    4. yes 
    5. yes

    Exercise \(\PageIndex{5}\)

    Determine whether each ordered pair is a solution to the inequality \(x+y>4\):

    1. \((5,1)\)
    2. \((−2,6)\)
    3. \((3,2)\)
    4. \((10,−5)\)
    5. \((0,0)\)

    Exercise \(\PageIndex{6}\)

    Determine whether each ordered pair is a solution to the inequality \(x+y>2\):

    1. \((1,1)\)
    2. \((4,−3)\)
    3. \((0,0)\)
    4. \((−8,12)\)
    5. \((3,0)\)
    Answer
    1. no 
    2. no 
    3. no 
    4. yes 
    5. yes

    Recognize the Relation Between the Solutions of an Inequality and its Graph

    In the following exercises, write the inequality shown by the shaded region.

    Exercise \(\PageIndex{7}\)

    Write the inequality shown by the graph with the boundary line \(y=3x−4\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3x minus 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the right of the line is shaded.

    Exercise \(\PageIndex{8}\)

    Write the inequality shown by the graph with the boundary line \(y=2x−4\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2x minus 4 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

    Answer

    \(y<2 x-4\)

    Exercise \(\PageIndex{9}\)

    Write the inequality shown by the graph with the boundary line \(y=\frac{1}{2} x+1\)

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-half x plus 1 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

    Exercise \(\PageIndex{10}\)

    Write the inequality shown by the graph with the boundary line \(y=-\frac{1}{3} x-2\)

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

    Answer

    \(y \leq-\frac{1}{3} x-2\)

    Exercise \(\PageIndex{11}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(x+y=5\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 5 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

    Exercise \(\PageIndex{12}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(x+y=3\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 3 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

    Answer

    \(x+y \geq 3\)

    Exercise \(\PageIndex{13}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(2x+y=−4\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus y equals negative 4 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

    Exercise \(\PageIndex{14}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(x+2y=−2\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 2 y equals negative 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

    Answer

    \(x+2 y \geq-2\)

    Exercise \(\PageIndex{15}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(3x−y=6\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

    Exercise \(\PageIndex{16}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(2x−y=4\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

    Answer

    \(2 x-y<4\)

    Exercise \(\PageIndex{17}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(2x−5y=10\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 5 y equals 10 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

    Exercise \(\PageIndex{18}\)

    Write the inequality shown by the shaded region in the graph with the boundary line \(4x−3y=12\).

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x minus 3 y equals 12 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

    Answer

    \(4 x-3 y>12\)

    Graph Linear Inequalities

    In the following exercises, graph each linear inequality.

    Exercise \(\PageIndex{19}\)

    Graph the linear inequality \(y>\frac{2}{3} x-1\)

    Exercise \(\PageIndex{20}\)

    Graph the linear inequality \(y<\frac{3}{5} x+2\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals three-fifths x plus 2 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

    Exercise \(\PageIndex{21}\)

    Graph the linear inequality \(y \leq-\frac{1}{2} x+4\)

    Exercise \(\PageIndex{22}\)

    Graph the linear inequality \(y \geq-\frac{1}{3} x-2\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

    Exercise \(\PageIndex{23}\)

    Graph the linear inequality \(x-y \leq 3\)

    Exercise \(\PageIndex{24}\)

    Graph the linear inequality \(x-y \geq -2\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 2 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

    Exercise \(\PageIndex{25}\)

    Graph the linear inequality \(4x+y>-4\)

    Exercise \(\PageIndex{26}\)

    Graph the linear inequality \(x+5y<-5\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 5 y equals negative 5 is plotted as a dashed line extending from the top left toward the bottom right. The region below the line is shaded.

    Exercise \(\PageIndex{27}\)

    Graph the linear inequality \(3 x+2 y \geq-6\)

    Exercise \(\PageIndex{28}\)

    Graph the linear inequality \(4 x+2 y \geq-8\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x plus 2 y equals negative 8 is plotted as a solid line extending from the top left toward the bottom right. The region to the right of the line is shaded.

    Exercise \(\PageIndex{29}\)

    Graph the linear inequality \(y>4x\)

    Exercise \(\PageIndex{30}\)

    Graph the linear inequality \(y>x\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x is plotted as a solid line extending from the bottom left toward the top right. The region above the line is shaded.

    Exercise \(\PageIndex{31}\)

    Graph the linear inequality \(y \leq-x\)

    Exercise \(\PageIndex{32}\)

    Graph the linear inequality \(y \leq-3 x\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a solid line extending from the top left toward the bottom right. The region to the left of the line is shaded.

    Exercise \(\PageIndex{33}\)

    Graph the linear inequality \(y \geq-2\)

    Exercise \(\PageIndex{34}\)

    Graph the linear inequality \(y<-1\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed horizontal line. The region below the line is shaded.

    Exercise \(\PageIndex{35}\)

    Graph the linear inequality \(y<4\)

    Exercise \(\PageIndex{36}\)

    Graph the linear inequality \(y \geq 2\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 is plotted as a solid horizontal line. The region above the line is shaded.

    Exercise \(\PageIndex{37}\)

    Graph the linear inequality \(x \leq 5\)

    Exercise \(\PageIndex{38}\)

    Graph the linear inequality \(x>-2\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 2 is plotted as a dashed vertical line. The region to the right of the line is shaded.

    Exercise \(\PageIndex{39}\)

    Graph the linear inequality \(x>-3\)

    Exercise \(\PageIndex{40}\)

    Graph the linear inequality \(x \leq 4\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals 4 is plotted as a solid vertical line. The region to the left of the line is shaded.

    Exercise \(\PageIndex{41}\)

    Graph the linear inequality \(x-y<4\)

    Exercise \(\PageIndex{42}\)

    Graph the linear inequality \(x-y<-3\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.

    Exercise \(\PageIndex{43}\)

    Graph the linear inequality \(y \geq \frac{3}{2} x\)

    Exercise \(\PageIndex{44}\)

    Graph the linear inequality \(y \leq \frac{5}{4} x\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-fourths x is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

    Exercise \(\PageIndex{45}\)

    Graph the linear inequality \(y>-2 x+1\)

    Exercise \(\PageIndex{46}\)

    Graph the linear inequality \(y<-3 x-4\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x minus 4 is plotted as a dashed line extending from the top left toward the bottom right. The region to the left of the line is shaded.

    Exercise \(\PageIndex{47}\)

    Graph the linear inequality \(x \leq-1\)

    Exercise \(\PageIndex{48}\)

    Graph the linear inequality \(x \geq 0\)

    Answer

    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 0 is plotted as a solid vertical line along the y-axis. The region to the right of the line is shaded.

    Everyday Math

    Exercise \(\PageIndex{49}\)

    Money. Gerry wants to have a maximum of $100 cash at the ticket booth when his church carnival opens. He will have $1 bills and $5 bills. If \(x\) is the number of $1 bills and \(y\) is the number of $5 bills, the inequality \(x+5y \leq 100\) models the situation.

    1. Graph the inequality.
    2. List three solutions to the inequality \(x+5y \leq 100\) where both \(x\) and \(y\) are integers.

    Exercise \(\PageIndex{50}\)

    Shopping. Tula has $20 to spend at the used book sale. Hardcover books cost $2 each and paperback books cost $0.50 each. If \(x\) is the number of hardcover books Tula can buy and \(y\) is the number of paperback books she can buy, the inequality \(2x+\frac{1}{2} y \leq 20\) models the situation.

    1. Graph the inequality.
    2. List three solutions to the inequality \(2x+\frac{1}{2} y \leq 20\) where both \(x\) and \(y\) are whole numbers.
    Answer

    1.
    The graph shows the x y-coordinate plane. The x- axis runs from 0 to 20 and the y-axis runs from 0 to 30. The line 2 x plus one-half y equals 20 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

    2. Answers will vary.

    Writing Exercises

    Exercise \(\PageIndex{51}\)

    Lester thinks that the solution of any inequality with a \(>\) sign is the region above the line and the solution of any inequality with a \(<\) sign is the region below the line. Is Lester correct? Explain why or why not.

    Exercise \(\PageIndex{52}\)

    Explain why in some graphs of linear inequalities the boundary line is solid but in other graphs it is dashed.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “verify solutions to an inequality in two variables,”, “recognize the relation between the solutions of an inequality and its graph,” and “graph linear inequalities.” The rest of the cells are blank.

    b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?