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Magazine Chapter 3: Problem 30
Graph each inequality. $$ 3 x+5 y \leq-2 $$
Short Answer
Expert verified
Shade the area below the line \(3x + 5y = -2\), where the line is solid.
Step by step solution
01
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.
02
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 \(-\frac{2}{5}\).
03
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.
04
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 \(3x + 5y \leq -2\)).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
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.
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.
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 \(0 ot\leq -2\).
