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3.4: Linear Inequalities in Two Variables

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    45041
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    Previously, we graphed inequalities in one variable, but now we learn to graph inequalities in two variables. Although this section may seem similar to linear equations in two variables, linear inequalities in two variables have many applications. For example, business owners want to know when revenue is greater than cost so that their business makes a profit, e.g., revenue \(>\) cost.

    Definition: Linear Inequality

    A linear inequality in two variables is an inequality of the form \[ax + by < c,\nonumber\] where the inequality is written in the same form for \(>,\: ≤,\: ≥\) and \(a,\: b\neq 0\).

    Recall. The solution to a linear inequality in one variable is an interval of numbers, e.g., \((−∞, ∞),\: [−2, 3),\: (1, 9),\: [−7, −3]\), etc.

    Verifying Solutions

    Solution to a Linear Inequality in Two Variables

    An ordered pair \((x, y)\) is a solution to a linear inequality in two variables, \(ax + by < c\), if the ordered pair \((x, y)\) makes the inequality true, where the the same is for \(>,\: ≤,\: ≥\) and \(a,\: b\neq 0\).

    Example \(\PageIndex{1}\)

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

    1. \((0,0)\)
    2. \((1,6)\)

    Solution

    We substitute the ordered pairs into the inequality and determine if the results are true.

    1. Let’s substitute \((0, 0)\) into the inequality and determine if the left side is greater than the right side. \[\begin{array}{rl}y\stackrel{?}{>}x+4&\text{Substitute }x=0\text{ and }y=0 \\ 0\stackrel{?}{>}0+4&\text{Simplify} \\ 0\cancel{>}4&X\text{ False}\end{array}\nonumber\] Hence, \((0,0)\) is not a solution to the inequality \(y>x+4\).
    2. Let’s substitute \((1, 6)\) into the inequality and determine if the left side is greater than the right side. \[\begin{array}{rl}y\stackrel{?}{>}x+4&\text{Substitute }x=1\text{ and }y=6 \\ 6\stackrel{?}{>}1+4&\text{Simplify} \\ 6>5&\checkmark\text{ True}\end{array}\nonumber\] Hence, \((1,6)\) is a solution to the inequality \(y>x+4\).

    Boundary Lines

    If we are given a linear inequality, \(ax + by < c\), we could see from Example \(\PageIndex{1}\) that not all ordered pairs are a solution, only some. Why? Well, notice that \((0, 0)\) is below the line \(y = x + 4\) and \((1, 6)\) is above the line \(y = x + 4\). This implies that ordered pairs in certain regions are solutions to the inequality \(y > x + 4\). Hence, the line \(y = x + 4\) is critical when finding solutions to the inequality. We call the line \(y = x + 4\) a boundary line, a line that separates the ordered pairs that are solutions and the ordered pairs that are not solutions of the linear inequality in two variables \(y > x + 4\).

    Definition: Boundary Line

    A linear equation in two variables \(ax + by = c\) is called the boundary line, the line that separates the region where \(ax + by > c\) and from the region where \(ax + by < c\).

    Note

    Since there are four inequality symbols: \(>,\: <,\: \geq ,\: \leq\), then we have linear inequalities in two variables that include the boundary, e.g., inequalities with \(\leq\) and \(\geq\), and linear inequalities in two variables that exclude the boundary, e.g., inequalities with \(<\) and \(>\).

    We can use the table below to help identify the boundary line, determine whether to include the boundary line, and the way the boundary line looks graphically.

    Table \(\PageIndex{1}\)

    Case 1. Case 2.
    \(ax+by<c\) \(ax+by\leq c\)
    \(ax+by>c\) \(ax+by\geq c\)
    Boundary line: \(ax+by=c\) Boundary line: \(ax+by=c\)
    Boundary line is excluded in solution Boundary line is included in solution
    Boundary line is dashed Boundary line is solid
    Example \(\PageIndex{2}\)

    Let’s revisit Example \(\PageIndex{1}\) and graph the boundary line and points \((0, 0)\) and \((1, 6)\).

    Solution

    Since we have \(y > x + 4\), we can see from the table above we have Case 1. and the boundary line is excluded. We represent this by graphing the line \(y = x + 4\) as a dashed line.

    clipboard_eecc28b0c216962c7fa170ca63f3324eb.png
    Figure \(\PageIndex{1}\)

    We can see from the graph that the point \((0, 0)\) lies below the boundary line \(y = x + 4\) and point \((1, 6)\) lies above the boundary line. Recall, point \((1, 6)\) was verified as a solution of \(y > x + 4\) in Example \(\PageIndex{1}\). Furthermore, any ordered pair that lies above \(y = x+ 4\) will verify as a solution, i.e., making the inequality true. We usually represent this area by shading the region where the set of ordered pairs make the inequality true.

    Graphing Linear Inequalities

    Example \(\PageIndex{3}\)

    Graph the inequality from Example \(\PageIndex{1}\).

    Solution

    Since we know that \((1, 6)\) is a solution to the inequality, then we shade above the dashed boundary line:

    clipboard_e08af2eba8fb540e91b042c9377fbba98.png
    Figure \(\PageIndex{2}\)

    We see that any ordered pair in the shaded region is a solution to the inequality. For example, let’s pick \((−4, 5)\) and verify this is a solution: \[\begin{array}{rl}y\stackrel{?}{>}x+4&\text{Substitute }x=-4\text{ and }y=5 \\ 5\stackrel{?}{>}-4+4&\text{Simplify} 5>0&\checkmark\text{ True}\end{array}\nonumber\]

    Hence, \((−4, 5)\) is a solution to the inequality \(y > x + 4\).

    Steps for Graphing Linear Inequalities in Two Variables

    Given a linear inequality in two variables, \(ax + by < c\), we use the steps below to graph \(ax + by < c\), where the the same process is applied for \(>,\: ≤,\: ≥\) and \(a,\: b\neq 0\).

    Step 1. Rewrite the inequality in slope-intercept form, i.e., \(y = mx + b\).

    Step 2. Graph the boundary line according to the two cases:

    Case 1. If the inequality is \(<\) or \(>\), then the boundary line is dashed.

    Case 2. If the inequality is \(≥\) or \(≤\), then the boundary line is solid.

    Step 3. Select a test point that is not on the boundary line. Ask: Does this ordered pair make the inequality true?

    Step 4. If the ordered pair is

    • a solution to the inequality, i.e., makes the inequality true, then shade the side that includes the ordered pair.
    • not a solution, then shade the opposite side of the boundary line.
    Note

    If we choose a test point on the boundary line, we will obtain an identity, where both sides of the inequality symbol are the same number. Hence, it is critical to choose a point not on the boundary line.

    Example \(\PageIndex{4}\)

    Graph the inequality: \[2x − y > 3\nonumber .\]

    Solution

    Let’s follow the steps given above to graph the inequality.

    Step 1. Rewrite the inequality in slope-intercept form, i.e., \(y = mx + b\). \[\begin{aligned} 2x-y&>3 \\ -y&>-2x+3 \\ y&<2x-3\end{aligned}\]

    Step 2. Graph the boundary line according to the two cases. Since the given inequality is \(<\), then we have Case 1.

    clipboard_ee3060257d515e142c73acafc38d4a63e.png
    Figure \(\PageIndex{3}\)

    Step 3. Select a test point that is not on the boundary line. Ask: Does this ordered pair make the inequality true?
    Let’s pick the test point \(\color{red}{(0, 0)}\) as it is a great choice! \[\begin{aligned} y&\stackrel{?}{<}2x-3 \\ 0&\stackrel{?}{<}2(0)-3 \\ 0&\cancel{\leq}-3\end{aligned}\] Hence, \((0,0)\) doesn't make the inequality true.

    Step 4. If the ordered pair is

    • a solution to the inequality, i.e., makes the inequality true, then shade the side that includes the ordered pair.
    • not a solution, then shade the opposite side of the boundary line.

    Since the ordered pair \((0, 0)\) is not a solution to the inequality, then we shade on the opposite side of the boundary line from the location of the ordered pair.

    clipboard_ee7c8fe9abb1880a9eb2ebc5e0bc4d9b2.png
    Figure \(\PageIndex{4}\)
    Note

    Another way of graphing linear inequalities in two variables is to complete Step 1. and Step 2., but instead of taking a test point in Step 3., we can observe the inequality symbols. If the inequality has \(<\) or \(≤\), then we easily shade below the boundary line, i.e., below the \(y\)-intercept. Similarly, if the inequality has \(>\) or \(≥\), then we easily shade above the boundary line, i.e., above the \(y\)-intercept.

    Example \(\PageIndex{5}\)

    Graph the inequality: \[3x + 2y ≥ −6.\nonumber\]

    Solution

    Let’s follow the steps given above to graph the inequality, but try skipping Step 3. and Step 4.

    Step 1. Rewrite the inequality in slope-intercept form, i.e., \(y = mx + b\). \[\begin{aligned}3x+2y&\geq -6 \\ 2y&\geq -3x-6 \\ y&\geq -\frac{3}{2}x-3\end{aligned}\]

    Step 2. Graph the boundary line according to the two cases. Since the given inequality is \(≥\), then we have Case 2.

    clipboard_e322356ef5503638339eee644521dc9c3.png
    Figure \(\PageIndex{5}\)

    Since this inequality is \(≥\), where all ordered pairs above the boundary line are solutions to the inequality, we can easily shade above the \(y\)-intercept:

    clipboard_e043d4129664dc9eb743361f01385e9d8.png
    Figure \(\PageIndex{6}\)

    Linear Inequalities in Two Variables Homework

    Determine whether the given ordered pairs are solutions to the inequality.

    Exercise \(\PageIndex{1}\)

    \(x + 2y ≥ −4;\: (0, −4);\: (1, 1)\)

    Exercise \(\PageIndex{2}\)

    \(2x − y ≤ 2;\: (1, 5);\: (3, 1)\)

    Graph the following inequalities.

    Exercise \(\PageIndex{3}\)

    \(2x − y ≤ 2\)

    Exercise \(\PageIndex{4}\)

    \(x > 4y − 8\)

    Exercise \(\PageIndex{5}\)

    \(x + 2y ≥ −4\)

    Exercise \(\PageIndex{6}\)

    \(3x + 4y < 12\)

    Exercise \(\PageIndex{7}\)

    \(6x + 8y ≤ 24\)

    Exercise \(\PageIndex{8}\)

    \(5x + 3y ≤ 15\)

    Exercise \(\PageIndex{9}\)

    \(y > 3x + 1\)

    Exercise \(\PageIndex{10}\)

    \(3x + 2y ≤ 12\)

    Exercise \(\PageIndex{11}\)

    \(5x − 2y < 10\)

    Exercise \(\PageIndex{12}\)

    \(3x + 4y ≥ 24\)

    Exercise \(\PageIndex{13}\)

    \(y ≤ 3x − 4\)


    This page titled 3.4: Linear Inequalities in Two Variables is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.