Problem 62 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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 1: 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: \(y \leq \frac{3}{4}x - 4\).

Step 2: 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.

Step 3: 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.

Key Concepts

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.