Problem 30 - Algebra A Combined Function

Question

Graph each inequality. $$ 3 x+5 y \leq-2 $$

Step-by-Step Solution

Answer

Shade the area below the line \(3x + 5y = -2\), where the line is solid.

Step 1: Rewrite the Inequality

We begin by rewriting the inequality in the format of a linear equation to clearly identify it as a line. The inequality is given as \(3x + 5y \leq -2\). We will consider \(3x + 5y = -2\) to find the boundary line.

Step 2: Determine Slope and Y-Intercept

Rewrite the boundary equation in slope-intercept form \(y = mx + b\). Start with \(3x + 5y = -2\). Solve for \(y\):Subtract \(3x\) from both sides: \[5y = -3x - 2\]Divide every term by 5: \[y = -\frac{3}{5}x - \frac{2}{5}\]The slope \(m\) is \(-\frac{3}{5}\) and the y-intercept is \(-\frac{2}{5}\).

Step 3: Graph the Boundary Line

Using the slope-intercept form, plot the y-intercept \((0, -\frac{2}{5})\) on the graph. From this point, use the slope \(-\frac{3}{5}\) to find another point. Move down 3 units and to the right 5 units to reach the second point \((5, -1)\). Draw a line through these points. Remember since the inequality is \(\leq\), this line will be solid.

Step 4: Determine the Shading Direction

Since the inequality is \(3x + 5y \leq -2\), shade the region below the line (this represents all the points \((x, y)\) that satisfy the inequality \(3x + 5y \leq -2\)).

Step 5: Verify with a Test Point

Select a test point not on the line to verify the shading. A common choice is the origin \((0, 0)\). Substitute into the inequality:\[3(0) + 5(0) \leq -2\rightarrow 0 \leq -2\]This is false, so the region containing \((0, 0)\) should not be shaded, confirming our earlier work.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a widely used format in linear equations. It is expressed as \(y = mx + b\) where \(m\) stands for the slope of the line and \(b\) represents the y-intercept. The slope \(m\) indicates the steepness and direction of the line, while the y-intercept \(b\) is the point where the line crosses the y-axis.
In the context of graphing inequalities, transforming the inequality into slope-intercept form helps us understand the characteristics of the boundary line.

• By rewriting the original inequality \(3x + 5y \leq -2\) as \(y = -\frac{3}{5}x - \frac{2}{5}\), we identify \(m = -\frac{3}{5}\) and \(b = -\frac{2}{5}\).
• This transformation allows you to swiftly graph the line by plotting the y-intercept and using the slope to find additional points.

To graph the line accurately, remember that the slope \(m = -\frac{3}{5}\) means you "go down 3 units" and "move right 5 units" from any point on the line.

Boundary Line

The boundary line is a crucial component when graphing inequalities. It represents the division between the solutions and non-solutions of the inequality.
In the example \(3x + 5y \leq -2\), the boundary line is derived by changing the inequality to an equation: \(3x + 5y = -2\).
This line is graphed to help visualize the area of interest.

• The boundary is depicted as a solid line if the inequality includes "equal to" (\(\leq\) or \(\geq\)), indicating that points on the line are solutions.
• If the inequality was strict (such as \(<\) or \(>\)), a dashed line would be used.

In our case, since the inequality is \(\leq\), we use a solid line. This indicates that all points on the line are included in the solution set.

Inequality Shading

When graphing linear inequalities, shading plays a vital role in identifying the solution region. After plotting the boundary line, the next step is to shade the appropriate side.
The inequality \(3x + 5y \leq -2\) tells us that we need to shade below the line since \(y\) needs to be "less than or equal" to the value on the line for any \(x\)-coordinate.

• Always start by assessing the direction of the inequality. The symbol \(\leq\) means shade below the boundary line.
• When shading, consider that the shaded region represents all the possible solutions to the inequality.

Remember, correct shading is essential to accurately show all the points that satisfy the inequality.

Test Point Method

The test point method is a helpful technique to verify the accuracy of shading on a graph. It's a simple strategy that involves choosing a point not on the boundary line and checking if it satisfies the inequality.

• A common test point used is the origin \((0, 0)\), because it often simplifies calculations unless the line passes through it.
• Substitute the test point into the inequality to see if it holds true. If true, the test point lies in the correct shaded area.
• In our example, from \(3x + 5y \leq -2\), substituting \((0,0)\) results in a false statement \(0 ot\leq -2\).

This confirms that the origin is not in the solution set, reinforcing that we've shaded the correct side of the boundary line.