Problem 20 - Intermediate Algebra

Question

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} 2 x-y \leq 4 \\ 2 x+y>4 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is the region below the line \( y = 2x - 4 \) and above the line \( y = -2x + 4 \).

Step 1: Graph the first inequality

First, consider the inequality \( 2x - y \leq 4 \). To graph this, you'll first rewrite it in slope-intercept form by solving for \( y \): \( y \geq 2x - 4 \). This represents a straight line where the region below or on the line is shaded. Plot the line by identifying two points: when \( x = 0 \), \( y = -4 \) and when \( y = 0 \), \( x = 2 \). Draw the line solid to indicate that the region on the line is included, and shade below the line.

Step 2: Graph the second inequality

For the inequality \( 2x + y > 4 \), rewrite it as \( y > -2x + 4 \). This time, plot the line by finding two points: when \( x = 0 \), \( y = 4 \) and when \( y = 0 \), \( x = 2 \). Since this is a strict inequality (\( > \)), draw a dashed line, indicating points on the line are not part of the solution. Shade above the line.

Step 3: Identify the solution set

The solution to the system of inequalities is the region that satisfies both inequalities simultaneously. This means looking for the area on the graph where the shaded regions from Steps 1 and 2 overlap. Ensure the overlapping region accounts for the conditions correctly: solid line from first inequality and dashed from the second indicate different inclusions.

Key Concepts

Graphing Inequalities

When graphing inequalities, you are essentially drawing lines on a coordinate plane to represent the constraints given by inequalities. Each inequality gets its own line, and different types of lines indicate different types of solutions. To graph an inequality, follow these steps:

• Convert the inequality into "y = mx + b" form, where "m" is the slope of the line and "b" is the y-intercept.
• If your inequality is "less than or equal to" (\(\leq\)), you'll draw a solid line. In this case, solutions are not just above or below the line, but also on the line itself.
• For a "greater than" (\(>\)) inequality, use a dashed line. This illustrates that solutions do not include the points on the line, only the region above or below it.

Once you've drawn your line, you must shade the correct side:

• For inequalities like \( y \leq mx + b \), shade below the line.
• For \( y \geq mx + b \), shade above the line.

Graphing in this way helps visually represent which values satisfy the inequality.

Solution Set

The solution set of a system of inequalities is the region on the graph where all shaded areas from the individual inequalities overlap. Only this intersection of shaded regions represents solutions that simultaneously satisfy each inequality in the system. Finding the solution set involves these steps:

• Plot each inequality on the same coordinate plane, following the graphing rules for solid or dashed lines as necessary.
• Shade the appropriate region for each inequality.
• Observe where the shaded sections overlap; this is your solution set.

This overlapping area must adhere to the different line rules:

• Dashed lines mean solutions hug but don't include the line.
• Solid lines include the line itself as well, within the solutions.

In our specific problem, carefully examining the lines (one solid and one dashed) ensures the correct solution set is highlighted.

Shaded Region

The shaded region plays a crucial role in visualizing the solutions to inequalities. By shading, we indicate which part of the graph includes numbers that satisfy the particular inequality.Here's how to think about shading:

• The direction of shading depends on whether the inequality is greater-than or less-than. More specifically, you'll shade above the line if the inequality involves greater-than (\(y > mx + b\) or \(y \geq mx + b\)), and shade below for less-than (\(y < mx + b\) or \(y \leq mx + b\)).
• Using distinct shading for different inequalities can make seeing the overlap clearer and more intuitive.
• In systems of inequalities, only the areas where all shadings from each inequality overlap will give valid solutions.

The nuances of shading and line style (solid vs. dashed) ensure a precise visual map of potential solutions, which is why it's critical to master these basics in graph-related problems.