Problem 30 - College Algebra

Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} 2 x-y \leq 4 \\ 3 x+2 y>-6 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution set of the given system of inequalities is the region that is shaded above both the solid line \(y = 2x - 4\) and the dashed line \(y = -3/2x + 3\). If the shaded regions do not overlap, the system has no solution.

Step 1: Rewrite Inequalities in Slope-Intercept Form

Start by rewriting the given inequalities into slope-intercept form. The first equation becomes \(y \geq 2x - 4\). Notice the inequality changes to ‘≥’ because we moved \(y\) to the other side. The second equation becomes \(y > -3/2x +3\).

Step 2: Plot the Lines

Now that both inequalities are in the form \(y = mx + b\), we plot the lines \(y = 2x - 4\) and \(y = -3/2x + 3\) on the same graph. Note that the line for the first inequality must be solid (since it includes ‘≥’), and the line for the second inequality must be dashed (since it’s just ‘>’, not ‘≥’). So, solid line for \(y = 2x - 4\) and dashed line for \(y = -3/2x + 3\).

Step 3: Shade the Inequalities

Now, for inequality \(y \geq 2x - 4\), since \(y\) is more than something, we shade the area above the line \(y = 2x - 4\). For inequality \(y > -3/2x + 3\), since \(y\) is more than something, we also shade above the line \(y = -3/2x + 3\).

Step 4: Identify the Solution Set

Look for the region where the shading from both inequalities overlaps. This area represents the solution set of the system of inequalities, which means all the points in this region satisfy both inequalities. If there's no overlapping region, that means the system has no solution.

Key Concepts

Slope-Intercept Form

When dealing with systems of inequalities, putting each equation into slope-intercept form can make graphing them easier. The slope-intercept form is written as \( y = mx + b \). Here, \( m \) represents the slope, which is the steepness of the line. The \( b \) is the y-intercept, which is where the line crosses the y-axis.
To convert an inequality into this form, isolate \( y \) on one side. For example, if you have \( 2x - y \leq 4 \), rearrange it to \( y \geq 2x - 4 \). By following this process, you can easily see how the line will look on a graph. Knowing the basics of slope-intercept form helps in graphing quickly and accurately.

Graphing Inequalities

Graphing inequalities involves drawing lines on the coordinate plane and determining the region of interest. Once your inequalities are in slope-intercept form, you can start drawing. Use the slope (\( m \)) and y-intercept (\( b \)) to plot your lines.
A key detail is recognizing which lines should be solid and which should be dashed. If the inequality symbol is 'greater than or equal to' (\( \geq \)) or 'less than or equal to' (\( \leq \)), use a solid line. This shows that points on the line are part of the solution. If the symbol is 'greater than' (\( > \)) or 'less than' (\( < \)), use a dashed line to show that points on the line are not included in the solution.

Solution Set

The solution set of a system of inequalities is the region where all conditions are satisfied. After graphing, you should identify where the shaded areas from each inequality overlap. This shared region represents all the possible solutions.
If there is no area where the shaded regions intersect, it means there is no solution that satisfies all inequalities simultaneously. It's crucial to review the entire graph to ensure no overlapping regions are missed. Understanding solution sets aids in visualizing how multiple conditions affect possible outcomes.

Inequality Shading

Shading is essential in graphing inequalities as it indicates where the solutions lie. For each inequality, determine which side of the line to shade. For cases like \( y \geq 2x - 4 \), shade above the line since \( y \) is greater than the expression. Similarly, for \( y > -3/2x + 3 \), shade above because the inequality specifies \( y \) is greater than the line equation.
Combining all shaded regions reveals the solution set. Always check from the line's perspective to ensure accurate shading. This step completes the visualization of the solution, making it easier to interpret which points satisfy the inequalities.