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Chapter 3: Problem 62

Graph each inequality on a coordinate plane. $$ 3 x-4 y \geq 16 $$

Short Answer

Expert verified
The shaded region is above the solid line \(y = \frac{3}{4}x - 4\), as the region including and above this line represents all the points that satisfy the inequality \(3x - 4y \geq 16\).

Step by step solution

01

Rewrite the inequality in slope-intercept form

Start by isolating the variable y on one side of the inequality to convert it into the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. First add 4y to both sides: \(3x \geq 4y + 16\). Then subtract 16 from both sides: \(3x - 16 \geq 4y\). Finally, divide by 4 across the inequality to solve for y: \(y \leq \frac{3}{4}x - 4\).
02

Graph the boundary line

The inequality includes the equal sign (\(\geq\)), so the boundary line will be solid. Graph the line \(\(y = \frac{3}{4}x - 4\) by first plotting the y-intercept, which is (0, -4), and then use the slope \(\)\frac{3}{4}\) to find another point. From the y-intercept, move up 3 units and right 4 units to plot the second point. Draw a solid line through these points, which is the boundary of the inequality.
03

Determine which side of the line to shade

Test a point that is not on the line to determine which half-plane to shade. A common point to use is the origin (0, 0), but since that lies on the boundary line (when y is made 0), choose a different point like (-1, 0). Substitute this point into the inequality: \(3(-1) - 4(0) \geq 16\). This simplifies to \(-3 \geq 16\), which is false. Since the point (-1, 0) does not satisfy the inequality, you shade the opposite side of the line, which is, in this case, the side that does not include the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
Understanding the slope-intercept form is essential when working with linear equations and inequalities. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

To transform an inequality into this form, just like in the textbook exercise example, isolate the \( y \) variable. Here's a clearer breakdown:
  • Add or subtract terms from both sides to gather the \( y \) terms on one side and the constants on the other.
  • Divide by the coefficient of \( y \) to solve for \( y \).
If the inequality sign is \( \geq \) or \( \leq \), it indicates that the boundary line is included in the solution. Conversely, if it's \( > \) or \( < \), the boundary line is not part of the solution.
Coordinate Plane
The coordinate plane is a two-dimensional surface formed by the intersection of two number lines: the horizontal x-axis and the vertical y-axis. Each point on the plane is represented by an ordered pair, \( (x, y) \).

Plotting a linear equation or inequality begins with identifying key points such as the y-intercept \( (0, b) \) and using the slope \( m \) to determine another point. The slope tells us how to move from one point to another on the line; a slope of \( \frac{3}{4} \) means that for every 3 units you go up (in the y-direction), you move 4 units to the right (in the x-direction).
Boundary Line
The boundary line in the context of graphing inequalities represents the division between the solutions that satisfy the inequality and those that do not.

Determining whether to draw a solid or dashed line is critical:
  • A solid line for the boundary is drawn when the inequality is \( \geq \) or \( \leq \), indicating that points on the line are included in the solution set.
  • A dashed line is used for \( > \) or \( < \), signifying that points on the line are not part of the solution.
As in the exercise, a solid boundary line for the inequality \( y \leq \frac{3}{4}x - 4 \) indicates that the line itself is part of the solution.
Shading Half-planes
After graphing the boundary line, the next step is to determine which side of the line contains the solutions to the inequality, known as shading the half-planes.

Here's a simple process for deciding where to shade:
  • Pick a test point not on the boundary line, often \( (0, 0) \) is a good choice unless it's on the line itself.
  • Substitute the coordinates of the test point into the original inequality.
  • If the inequality holds true, shade the side of the plane where the test point lies. If not, shade the opposite side.
In our exercise, since the point \( (-1, 0) \) does not satisfy \( 3x - 4y \geq 16 \), we shade the side opposite to where \( (-1, 0) \) lies.

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Most popular questions from this chapter

Solve each system by elimination. Check your answers. $$ \left\\{\begin{array}{l}{6 q-r+2 s=8} \\ {2 q+3 r-s=-9} \\ {4 q+2 r+5 s=1}\end{array}\right. $$

Solve each system. $$ \left\\{\begin{array}{l}{4 y+2 x=6-3 z} \\ {x+z-2 y=-5} \\ {x-2 z=3 y-7}\end{array}\right. $$

Solve each system by elimination or substitution. $$ \left\\{\begin{array}{l}{2=4 y-3 x} \\ {5 x=2 y-3}\end{array}\right. $$

Geometry In the regular polyhedron described below, all faces are congruent polygons. Use a system of three linear equations to find the numbers of vertices, edges, and faces. Every face has five edges and every edge is shared by two faces. Every face has five vertices and every vertex is shared by three faces. The sum of the number of vertices and faces is two more than the number of edges.

Solve each system. $$ \left\\{\begin{aligned} 3 x+2 y-z &=17.8 \\ x-3 y+2 z &=7.9 \\ 2 x+y-3 z &=3.9 \end{aligned}\right. $$
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