Problem 32 - Intermediate Algebra

Question

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} 2 x+5 y>10 \\ 5 x+2 y>10 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is the region above both dashed lines where they overlap.

Step 1: Understand the system of inequalities

We have a system of inequalities: \( 2x + 5y > 10 \) and \( 5x + 2y > 10 \). We need to determine the solution set by identifying the region that satisfies both inequalities.

Step 2: Rearrange the inequalities

First, rearrange each inequality to the slope-intercept form, \( y = mx + b \).- For \( 2x + 5y > 10 \): \[ 5y > -2x + 10 \] \[ y > -\frac{2}{5}x + 2 \]- For \( 5x + 2y > 10 \): \[ 2y > -5x + 10 \] \[ y > -\frac{5}{2}x + 5 \]

Step 3: Graph the lines

Graph the boundary lines for the inequalities. - The line for \( y = -\frac{2}{5}x + 2 \) is a dashed line since the inequality is '>'.- The line for \( y = -\frac{5}{2}x + 5 \) is also a dashed line.These lines represent the boundaries where equality holds, but the solution is the region where y is greater than these lines.

Step 4: Shade the regions

Shade the region above each boundary line since both inequalities have '>'.- Shade the region above the line \( y = -\frac{2}{5}x + 2 \).- Shade the region above the line \( y = -\frac{5}{2}x + 5 \).

Step 5: Identify the solution set

The solution set is where the shaded regions from both inequalities overlap. This is the region that satisfies both \( 2x + 5y > 10 \) and \( 5x + 2y > 10 \). Ensure both conditions are met in the overlapped region.

Key Concepts

Graphing Inequalities

Graphing inequalities is a fundamental skill in algebra that helps you visualize solutions for a system of inequalities. Instead of simple lines, inequalities describe regions on a graph. To graph inequalities, you follow these steps:

• First, treat the inequality as an equation to find the corresponding boundary line.
• Graph the boundary line using appropriate values. If the inequality symbol is ">" or "<", the line is dashed, indicating that points on the line are not included in the solution.
• After graphing the line, you will shade one side of it. The side to shade is determined by testing a point on the graph. Commonly, the origin (0,0) is a quick test point, unless it is directly on the boundary line.

Understanding these concepts helps you see where the solutions have taken form visually on the graph.

Solution Set

The solution set of a system of inequalities consists of all points that make all the inequalities in the system true. In the context of graphing, the solution set is represented as an overlapping shaded region on the graph.
Identifying the solution set involves:

• Plotting each inequality on the same graph to find their respective shaded regions.
• Finding the intersection or overlap of these shaded regions.

The intersecting area satisfies all inequalities simultaneously, meaning any point within this region will solve all the inequalities in the system.

Slope-Intercept Form

The slope-intercept form of a linear equation is essential for graphing. It is given by the formula:\[ y = mx + b \]where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

Rewriting inequalities in slope-intercept form makes it easy to plot the graph. You can easily identify how steep the line is and where it crosses the y-axis, allowing a quick sketch of the boundary line.

• Convert each inequality by first isolating The terms of the inequality.
• Apply algebra to rearrange into the form \( y = mx + b \), keeping inequality direction in mind.

Graphing begins once the line is fully identified using this form.

Shading Regions

Shading regions on the graph demonstrates where different inequalities are true. The shaded part shows the domain where the solutions of the inequality satisfy the condition specified.
Once you graph the boundary line of the inequality, you'll:

• Determine which side of the line to shade by testing a point.
• If the test point satisfies the inequality, shade the region containing that point. If not, shade the opposite side.
• Repeat this method for all inequalities in the system.

The combined shaded region from all inequalities is where they intersect, representing the solution set for the whole system. Be certain to pay attention to whether lines are dashed or solid as this impacts the shading decisions.