Problem 55 - Elementary Algebra

Question

$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ 3 x+4 y>-12 $$

Step-by-Step Solution

Answer

Shade the region above the dashed line through (-4, 0) and (0, -3).

Step 1: Rewrite the Inequality as an Equation

To graph the inequality, first rewrite it as an equation for convenience: \( 3x + 4y = -12 \). This will help in determining the boundary line, which we'll draw on the graph.

Step 2: Find the Intercepts

To graph the line, find the x- and y-intercepts of the equation \( 3x + 4y = -12 \). For the x-intercept, set \( y = 0 \) and solve for \( x \): \( 3x = -12 \Rightarrow x = -4 \). For the y-intercept, set \( x = 0 \) and solve for \( y \): \( 4y = -12 \Rightarrow y = -3 \). This gives intercepts at (-4, 0) and (0, -3).

Step 3: Draw the Boundary Line

Plot the intercept points (-4, 0) and (0, -3) on the graph. Connect these points with a straight line. Since the original inequality is \( 3x + 4y > -12 \) (note: strictly greater than), draw the line as dashed to indicate that points on the line itself are not included in the solution.

Step 4: Choose a Test Point

Select a test point not on the boundary line to determine which side of the line to shade. A simple choice is the origin (0, 0), unless it lies on the line. Substitute into the inequality: \( 3(0) + 4(0) > -12 \) simplifies to \( 0 > -12 \), which is true.

Step 5: Shade the Solution Region

Since the test point (0, 0) satisfies the inequality, shade the region of the graph that includes the origin. This shaded area represents all the points \( (x, y) \) that satisfy \( 3x + 4y > -12 \).

Key Concepts

Boundary Line

The boundary line is a crucial part of graphing linear inequalities. In this case, we start by changing the inequality into an equation: \(3x + 4y = -12\). This line acts as a divider, separating the graph into different regions. To plot the boundary line, first find the intercepts which give you an idea of where the line crosses the axes. The boundary line here is drawn as dashed, not solid, because the inequality \(3x + 4y > -12\) does not include the line itself. The line being dashed visually represents that points on it satisfy the equation, not the inequality.

X-Intercept

To determine the x-intercept, we focus on the point where the boundary line crosses the x-axis. This occurs when the value of \(y\) is zero. For the equation \(3x + 4y = -12\), we set \(y = 0\) and solve for \(x\):

• \(3x = -12\)
• \(x = -4\)

Thus, the x-intercept is at the point \((-4, 0)\). It means that at this point, the line touches the x-axis, giving a starting spot to draw the boundary line on the graph.

Y-Intercept

The y-intercept is the point where the boundary line crosses the y-axis, found by setting \(x = 0\) in the equation. This will clarify how the line behaves as it crosses that axis. For our equation \(3x + 4y = -12\):

• \(4y = -12\)
• \(y = -3\)

This calculation provides the y-intercept at \((0, -3)\). Both intercepts, x and y, are necessary to accurately draw the boundary line on the graph. These intercepts offer precise points to ensure the drawn line is correct.

Test Point

A test point helps to identify which side of the boundary line to shade. It's crucial to choose a point not lying on the line. Commonly, the origin \((0, 0)\) is a simple and effective choice unless it is exactly on the line. By substituting the test point into the inequality \(3x + 4y > -12\), if the inequality holds true, that implies the region containing the test point should be shaded.

• Here, substitute \((0, 0)\):
• \(3(0) + 4(0) > -12\)
• Which gives \(0 > -12\), a true statement.

Therefore, the region containing the origin is where the inequality holds, pointing to the side of the line that requires shading.

Shading Solution Region

Shading indicates all solutions to the inequality. Once the correct side of the boundary line is determined using the test point, proceed to shade this region. In our case with \(3x + 4y > -12\), since the origin satisfies the inequality, you shade the half-plane that includes this point.The shaded area effectively communicates all coordinates \((x, y)\) that solve the inequality. This visualization is important; where the boundary line divides the graph, one clearly sees which region is part of the solution. This enables you to immediately identify which points satisfy the inequality just by glancing at the graph.