Problem 299 - Elementary Algebra

Question

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \geq \frac{3}{4} x-2 \\ y<2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the overlapping shaded region above the line \( y = \frac{3}{4}x - 2 \) and below the line \( y = 2 \).

Step 1 - Graph the First Inequality

Start by graphing the boundary line for the inequality \( y = \frac{3}{4}x - 2 \). This line has a slope of \( \frac{3}{4} \) and a y-intercept at \( -2 \). Since the inequality is \( y \geq \frac{3}{4}x - 2 \), shade the region above the line.

Step 2 - Graph the Second Inequality

Next, graph the boundary line for the inequality \( y = 2 \). This is a horizontal line that crosses the y-axis at \( y = 2 \). Because the inequality is \( y < 2 \), shade the region below this line.

Step 3 - Find the Intersection Region

Look at the areas shaded in both Step 1 and Step 2. The solution to the system of inequalities will be the region where the shaded areas overlap. This region represents all the points that satisfy both inequalities.

Step 4 - Verify Solution

Verify the intersection region by selecting a sample point within the overlapping area and checking that it satisfies both inequalities. For example, if the point (0, 0) is in the intersection region, verify it satisfies \( 0 \geq \frac{3}{4}(0) - 2 \) and \( 0 < 2 \).

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting a line and then shading a region to show where the inequality holds. Start with the given inequality, convert it to its boundary line, and plot. For instance, consider the inequality: \( y \geq \frac{3}{4}x - 2 \. \) Firstly, the boundary line here is \( y = \frac{3}{4}x - 2 \). Determine the slope \( \left(\frac{3}{4}\right) \) and y-intercept \( -2 \). Plot a few points using these two. If the inequality includes equal to \( \leq, \geq \) use a solid line. If not \( <, > \), use a dashed line. Then, shade the region above or below this line based on the inequality's sign. This shading indicates the solutions for that inequality.

Intersection of Inequalities

Once you have graphed both inequalities, the next step is finding their intersection. The intersection is the region where the shaded areas from both inequalities overlap. This region contains points that satisfy both inequalities simultaneously. For example, after graphing \( y \geq \frac{3}{4}x - 2\ \) and \( y < 2 \) separately, you notice a common area where the shading overlaps. This overlapping section is the solution to the system of inequalities. It is where the conditions from both inequalities are true. Check this easily by selecting a point from the intersection and ensuring it works for both inequalities.

Boundary Lines

Boundary lines are the straight lines you graph first, representing the equality part of the inequality. For example, if you start with \( y = \frac{3}{4}x - 2\ \), this is a straight line with slope \( \frac{3}{4} \) and y-intercept \( -2\). Graphing boundary lines involves plotting points using the slope and intercept. Decide whether the boundary line should be solid or dashed; solid lines are for \( \leq \) or \( \geq \), while dashed lines are for \( < \) or \( > \). After drawing this line, you're ready to shade the appropriate region. The second line is \( y = 2\ \), a horizontal line crossing the y-axis at 2. Depending on the inequality sign, shade above or below.

Shading Regions in Graph

Shading the regions in a graph visually represents where the inequality holds true. Once the boundary line is in place, look at the inequality sign. For \( y \geq \frac{3}{4}x - 2 \), you will shade above the line, indicating all points above the line meet the inequality condition. For \( y < 2 \), you shade below the horizontal line at y = 2. The shaded areas show all the possible solutions for each inequality. The intersection or common shaded area of both inequalities then gives the solution to the system. Ensure clarity by shading distinctly and checking the intersection’s points.