Problem 9 - Algebra for College Students

Question

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{r} x+2 y>-2 \\ x-y<-3 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is the region above the line \( x + 2y = -2 \) and below the line \( x - y = -3 \).

Step 1: Identify Inequalities

We have two inequalities: \( x + 2y > -2 \) and \( x - y < -3 \). Our goal is to find and graph the solution set for these inequalities.

Step 2: Convert to Equations

Convert each inequality into an equation by replacing the inequality sign with an equal sign: \( x + 2y = -2 \) and \( x - y = -3 \). We will use these equations to find the boundary lines.

Step 3: Graph the Boundary Lines

Graph the lines represented by the equations from Step 2. For the line \( x + 2y = -2 \), rearrange into slope-intercept form as \( y = -\frac{1}{2}x - 1 \). For the line \( x - y = -3 \), rearrange as \( y = x + 3 \). Plot these lines on the coordinate plane.

Step 4: Test Points to Determine Regions

Select a test point that is not on either line (commonly the origin \((0,0)\) if not on any line) and substitute it into the original inequalities to determine which side of each line satisfies the inequality. In \( x + 2y > -2 \), using \((0,0)\) gives \( 0 + 0 > -2 \), which is true. In \( x - y < -3 \), using \((0,0)\) gives \( 0 - 0 < -3 \), which is false.

Step 5: Shade the Correct Region

For the inequality \( x + 2y > -2 \), shade the region above the line (the side containing the point \((0,0)\)). For \( x - y < -3 \), shade the region below the line (the side not containing \((0,0)\)). The solution set is the intersection of both shaded regions.

Key Concepts

Graphing Inequalities

When dealing with systems of inequalities, one common method is to graph each inequality on the same coordinate plane. This helps visualize the different regions defined by the inequalities. First, express each inequality in slope-intercept form, which is the form \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. This adaptation makes it straightforward to plot the line.

• The first inequality, \( x + 2y > -2 \), is converted to \( y = -\frac{1}{2}x - 1 \).
• The second inequality, \( x - y < -3 \), becomes \( y = x + 3 \).

Plotting these lines will involve finding two points on each line and drawing a straight line through them. It is crucial to pay attention to the type of line used: dashed lines are used for \( > \) or \( < \) inequalities, indicating that points on the line are not included in the solution set.

Solution Set

Once both inequalities are graphed, the solution set is the region where both conditions are true simultaneously. This is found by determining which part of the graph is shared by both inequalities.
With shading, the graph visually illustrates where the solutions overlap—this overlap area represents all the values of \( x \) and \( y \) that satisfy both inequalities. Always remember:

• The solution set of a system of inequalities is the intersection of the shaded regions.
• It’s helpful to clearly label these shaded regions to avoid confusion.

Check the solution set by picking any point within the overlapping area, if the point satisfies both inequalities, then the graph is likely correct.

Boundary Lines

In graphing a system of inequalities, the boundary lines serve as the dividing lines that separate various regions of potential solutions. These lines are crucial for establishing where one should start shading.

• The boundary line for \( x + 2y = -2 \) is drawn from points such as \( (0, -1) \) and another point after determining with the slope.
• The boundary line for \( x - y = -3 \) uses points like \( (0, 3) \).

These give you straight lines that provide a reference to determine which side of the line includes the solution set. If the inequality is \( > \) or \( < \), use a dashed line to show that the points on the line are not included.In contrast, if the inequality is \( \geq \) or \( \leq \), a solid line indicates that points on the line are part of the solution set.

Test Points

Selecting test points helps confirm which side of a boundary line includes solutions to an inequality. It’s usually easiest to use the origin, \( (0, 0) \), unless it lies on the boundary line itself.
Here’s how this works:

• Plug the test point into the original inequality. For the first inequality, \( x + 2y > -2 \) with \( (0, 0) \) gives \( 0 > -2 \), which holds true, indicating that \( (0, 0) \) is in the solution region of this inequality.
• For the second inequality, \( x - y < -3 \), the same point \((0, 0)\) provides \( 0 < -3 \), which is false, suggesting \( (0, 0) \) is not part of this solution region.

Choosing test points and evaluating them against each inequality ensures the shaded area accurately represents all solutions to the system of inequalities. It’s a vital step to verify the correctness of your graph.