Problem 28 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&y<-2 x+4\\\&y

Step-by-Step Solution

Answer

The solution set to the given system of inequalities is found where the shaded regions of the inequalities overlap, as determined by graphing each inequality and testing a point not on either line. If there is no region where the shading overlaps, the system has no solution.

Step 1: Graph the first inequality

Start with graphing the line of the first inequality \(y = -2x + 4\). This line has a slope of -2 and a y-intercept of 4. Because our original inequality is less than (not less than or equal), the line should be dashed, not solid. Choose a point not on the line, usually (0,0) is an easy choice, unless our line goes through that point. Substitute the chosen point into the inequality. If it makes the inequality true, shade that side of the line. If it does not make the inequality true, shade the other side of the line.

Step 2: Graph the second inequality

Same strategy is used for the second inequality \(y = x - 4\). This line has a slope of 1 and a y-intercept of -4. The line should be dashed. Choose a test point not on the line, preferably (0,0), substitute into the inequality. If it makes the inequality true, shade the corresponding side of the line, otherwise shade the opposite side.

Step 3: Define the solution set

The solution set of the system of inequalities is the area where the shaded regions for each of the two inequalities overlap. If there is no such area, then the system has no solution.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing regions on a coordinate plane that satisfy certain mathematical conditions. When working with inequalities such as linear inequalities, you begin by graphing the corresponding equation as if it were an equation of a line. This line acts as a boundary for the inequality. The visual representation helps see where the inequality holds true.

• Identify the inequality equation, for example, let's use an inequality like \( y < -2x + 4 \).
• Convert the inequality to an equation: \( y = -2x + 4 \).
• Graph this line. If the inequality is strictly less than (<) or greater than (>), use a dashed line, indicating that points on the line are not included in the solution.
• Choose a test point not on the line, like (0,0), and plug it into the inequality to check which side of the line to shade. If the resulting statement is true, shade the side containing the test point.

The shaded region represents where the inequality statement holds true. Follow these steps for each inequality in the system.

Solution Set

The solution set in a system of inequalities refers to the set of all possible solutions that satisfy all inequalities simultaneously. When graphing systems of linear inequalities, finding the solution set combines graphical analysis and logical intersections of shaded regions.

• Begin by graphing each individual inequality on the same coordinate plane, following the steps outlined for graphing inequalities.
• After graphing, identify the common shaded region, which represents all points that satisfy every inequality in the system.
• The intersection area of all shaded regions on your graph is the solution set.

If there is no overlapping area in the graph, the system of inequalities has no solution. Here, for example, unless a third inequality or other constraints provide intersections, we simply look for where both conditions are true. Understanding where these conditions are met graphically is critical.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line. It makes graphing simpler and helps in identifying the slope and y-intercept of the line quickly. This form is very useful when dealing with linear inequalities.

• The general form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• The slope \( m \) indicates the steepness and direction of the line - a positive slope moves upwards to the right, and a negative slope downwards.
• The y-intercept \( b \) is the point where the line crosses the y-axis.

For example, in the exercise above, \( y = -2x + 4 \) has a slope of -2 and intercept of 4, guiding you to draw the line starting at (0,4) and descending 2 units down for 1 unit across on the x-axis. Knowing this form allows you to efficiently graph endpoints and understand inequalities visually. This understanding is essential in accurately representing the regions associated with each inequality.