Problem 29 - Algebra 2

Question

Graph each inequality. $$ 4 x-5 y-10 \leq 0 $$

Step-by-Step Solution

Answer

Graph a solid line for \(y = \frac{4}{5}x - 2\) and shade above it.

Step 1: Write the Inequality in Slope-Intercept Form

The given inequality is \(4x - 5y - 10 \leq 0\). First, we want to rewrite this inequality in the form \(y = mx + b\) to make graphing easier. Begin by isolating \(y\): \[ 4x - 5y \leq 10 \]Subtract \(4x\) from both sides:\[-5y \leq -4x + 10 \] Now, divide every term by \(-5\). Remember, dividing an inequality by a negative number reverses the inequality sign:\[ y \geq \frac{4}{5}x - 2 \].

Step 2: Graph the Boundary Line

Now we will graph the boundary line of the inequality \(y = \frac{4}{5}x - 2\). Since the inequality is \(\geq\), use a solid line indicating that points on the line satisfy the inequality. 1. Start by plotting the y-intercept \((0, -2)\).2. From the y-intercept, use the slope \(\frac{4}{5}\): rise 4 units up and run 5 units to the right to locate another point and plot it.3. Draw the solid line through these two points.

Step 3: Shade the Solution Region

Next, determine which side of the line to shade. Pick a test point that is not on the line, such as the origin \((0,0)\), and substitute it into the inequality:\[ 0 \geq \frac{4}{5}(0) - 2\]This simplifies to \(0 \geq -2\), which is true. Thus, the region including the origin is the solution region. Shade this region on the graph.

Step 4: Verify the Solution

Finally, verify the solution by checking if the shaded region satisfies the original inequality. Any point in the shaded area should make \(4x - 5y - 10 \leq 0\) true. For example, check point \((5,0)\):4(5) - 5(0) - 10 = 20 - 10 = 10,Which satisfies \(10 \leq 10\). The solution is verified.

Key Concepts

Slope-Intercept Form

Rewriting an inequality in the slope-intercept form, which is represented by the equation \(y = mx + b\), is an essential step in graphing it. This form allows you to easily identify the slope \(m\) and the y-intercept \(b\) of the line. For the inequality \(4x - 5y - 10 \leq 0\), we start by solving for \(y\). This process involves moving terms around until \(y\) is on one side by itself.
The steps are as follows:

• Start with the original inequality: \(4x - 5y - 10 \leq 0\).
• Move the \(4x\) to the other side by subtracting \(4x\) from both sides: \(-5y \leq -4x + 10\).
• Divide each term by \(-5\) to isolate \(y\). Remember, dividing an inequality by a negative number reverses the inequality sign: \(y \geq \frac{4}{5}x - 2\).

Now the inequality is ready to be graphed with a slope of \(\frac{4}{5}\) and a y-intercept of \(-2\).

Boundary Line

Once the inequality is in slope-intercept form, the next step is to graph its boundary line. The concept of a boundary line is crucial because it shows the line where the inequality "changes". In the inequality \(y \geq \frac{4}{5}x - 2\), the boundary line is \(y = \frac{4}{5}x - 2\).
Here's how you graph it:

• First, identify the y-intercept, which is the point where the line crosses the y-axis. In this case, it's \((0, -2)\).
• Next, use the slope \(\frac{4}{5}\) to determine the direction and steepness of the line. Start at the y-intercept and move up 4 units (rise) and 5 units to the right (run) to plot another point.
• Connect these points with a solid line because the inequality \(y \geq\) includes equality (points on the line are included).

This solid line represents all the points where \(y\) is exactly \(\frac{4}{5}x - 2\).

Solution Region Shading

After plotting the boundary line, the next challenge is to determine which side of the line represents the solutions to the inequality. This process involves shading the solution region on the graph.
To do this, you:

• Select a test point that is not on the boundary line. The origin \((0, 0)\) is a convenient choice if it isn’t on the line.
• Substitute the test point into the inequality \(y \geq \frac{4}{5}x - 2\).
• Evaluate the inequality using the test point. If the inequality holds true, shade the side of the line where the test point is located.

For the given inequality, testing \((0, 0)\) yields: \(0 \geq -2\), which is true. This indicates that the side of the line including the origin is the solution region, which should be shaded completely.

Inequality Verification

Verifying the solution is an essential final step. It ensures that the shaded region correctly represents solutions to the inequality. Here's how to verify it:

• Select a few points within the shaded region and substitute them into the original inequality \(4x - 5y - 10 \leq 0\).
• If each point satisfies the inequality, the shading is correct.

In this example, using point \((5, 0)\):

• Substitute into the inequality: \(4(5) - 5(0) - 10 = 10\).
• Check if the inequality holds: \(10 \leq 10\), which is true.

If all tested points meet the condition, your graphing is accurate and complete.