Problem 300 - Elementary Algebra

Question

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \leq-\frac{1}{2} x+3 \\ y<1 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the lines, shade the regions, and find the overlap.

Step 1 - Graph the first inequality

The first inequality is \( y \leq -\frac{1}{2}x + 3 \). Rewrite it as the equation \( y = -\frac{1}{2}x + 3 \) to graph the boundary line. Plot the y-intercept (0, 3) and the slope \( -\frac{1}{2} \). From the y-intercept, move down 1 unit and right 2 units to find another point. Draw a solid line through these points because the inequality includes 'equal to'. Shade the region below this line.

Step 2 - Graph the second inequality

The second inequality is \( y < 1 \). Rewrite it as the equation \( y = 1 \) to graph the boundary line. Plot the horizontal line passing through y = 1. Draw a dashed line because the inequality does not include 'equal to'. Shade the region below this line.

Step 3 - Find the solution region

The solution to the system is the overlap of the shaded regions from both inequalities. The solution region is where the two shaded regions intersect.

Key Concepts

graphing inequalities

To solve systems of inequalities by graphing, you need to visualize each inequality on a coordinate plane. Start by turning the inequality into an equation. For example, for the inequality \( y \leq -\frac{1}{2}x + 3 \), you first graph the equation \( y = -\frac{1}{2}x + 3 \).

Find key points using the slope and y-intercept. Here, the y-intercept is (0,3) and the slope is -\frac{1}{2}, which means you move down 1 unit and to the right 2 units. Draw a line through these points.

The type of line depends on the inequality symbol:

• Use a solid line if the inequality includes 'equal to' (≤ or ≥).
• Use a dashed line if it doesn’t include 'equal to' (< or >).

Next, shade the area where the inequality holds true. For the inequality \( y \leq -\frac{1}{2}x + 3 \), you shade below the line.

solution region

The solution region of a system of inequalities is where the shaded areas of the individual inequalities overlap. Each inequality restricts part of the plane, and the solution satisfies all inequalities simultaneously.

After graphing each inequality and shading their respective regions, identify the common area shared by all inequalities.

If there is no such region, then the system of inequalities has no solution. Otherwise, the overlapping region is the solution region, representing all possible solutions. To solidify the understanding, always double-check by picking a point in the solution area and verifying it satisfies all given inequalities.

boundary lines

Boundary lines are crucial in graphing inequalities because they define the edges of the solution regions. For each inequality, the boundary line can either be solid or dashed, based on the inequality symbol.

• Solve the inequality to standard form (like \( y = mx + b \)) to plot the boundary line.
• Solid lines are used for inequalities with ≤ or ≥.
• Dashed lines are used for strict inequalities (< or >).

For example, to graph \( y < 1 \), draw a dashed line along y = 1 because 1 is not included in the solutions. Always make sure to shade correctly relative to these lines to denote the valid area for the inequality.

Never forget, boundary lines help in visualizing the dividing line between solutions that satisfy the inequality and those that do not.