Problem 36 - Algebra 2

Question

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x \leq 1} \\ {y < 2 x+1} \\ {x+2 y \geq-3}\end{array} $$

Step-by-Step Solution

Answer

The solution is the region where the shaded areas from all three inequalities overlap.

Step 1: Graph the Boundary Line for \(x \leq 1\)

Draw the vertical line \(x = 1\) on the coordinate plane. This line represents the boundary of the inequality \(x \leq 1\). The region to the left of this line, including the line itself, is the solution region for this inequality. Shade this region lightly.

Step 2: Graph the Boundary Line for \(y < 2x + 1\)

First, graph the line \(y = 2x + 1\). Since the inequality is a strict inequality (\(<\)), use a dashed line to represent \(y = 2x + 1\). Shade the region below this line, as that represents \(y < 2x + 1\).

Step 3: Graph the Boundary Line for \(x + 2y \geq -3\)

Convert the inequality into the equation \(x + 2y = -3\). Re-arrange to slope-intercept form: \(y = -\frac{1}{2}x - \frac{3}{2}\). Draw a solid line for \(y = -\frac{1}{2}x - \frac{3}{2}\), since the inequality is \(\geq\), including the boundary. Shade the region above this line.

Step 4: Identify the Feasible Region

The solution to the system of inequalities is the overlapping region that satisfies all three inequalities. This region should be the intersection of the shaded regions from the three previous steps.

Step 5: Verify a Point in the Solution Region

Choose a point from the intersection region, such as \((0,0)\), and substitute into each inequality to ensure it satisfies all of them: \(0 \leq 1\), \(0 < 2 \times 0 + 1 = 1\) and \(0 + 2(0) \geq -3\). Since these are all true, this point verifies the solution region.

Key Concepts

Graphing Inequalities

Graphing inequalities involves a few key steps to visualize solutions effectively. First, turn the inequality into an equation to get the boundary line. For example, with the inequality \(x \leq 1\), you simplify it to a line by writing \(x = 1\). Next, draw this line on a coordinate plot.

It is essential to decide if this line should be solid or dashed. If the inequality is "less than or equal to" (\(\leq\)) or "greater than or equal to" (\(\geq\)), use a solid line. This includes the points on the line as part of the solution. However, if it’s just "less than" (\(<\)) or "greater than" (\(>\)), use a dashed line. This indicates that the points on the line are not part of the solution.

Take the inequality \(y < 2x + 1\). Transform it into \(y = 2x + 1\) for graphing and use a dashed line because it’s a strict inequality.

• Change inequality to equation for graphing.
• Solid line for \(\leq\) or \(\geq\), dashed for \(<\) or \(>\).
• Shade the correct side representing the inequality.

Feasible Region

The feasible region is vital for systems of inequalities because it shows all the solution possibilities.

This region is found where the shaded areas of all included inequalities overlap. For example, after plotting the inequalities \((x \leq 1, y < 2x + 1, x + 2y \geq -3)\), the feasible region is the intersection zone of the respective shaded areas. It represents all the points that satisfy every inequality at the same time.

To find and confirm this area, test a point within the overlapped region. This point should satisfy all original inequalities. If it does, it's within the correct feasible region. As a result, visually checking and calculating with test points ensure accurate graphing of the feasible region.

• Represents all solutions of the system.
• Formed by overlapping shaded regions.
• Verify with test points from the region.

Boundary Lines

Boundary lines come directly from inequalities and are essential in determining solution areas on a graph.

First, identify boundary equations by altering inequalities. For example, \(x + 2y \geq -3\) turns into the equation \(x + 2y = -3\). This line is plotted as solid because the inequality includes equality (\(\geq\)).

These lines separate the coordinate plane into different regions. Once plotted, they tell you where to shade. For each line, determine if above or below it holds true for the inequality conditions properly. Such decisions revolve around whether the points on the line are to be included in the solution set or not.

Recognizing and accurately drawing boundary lines help in forming correct shaded areas. This not only ensures proper illustration of individual inequalities but also sets a foundation for understanding feasible regions.

• Convert inequality to equation for plotting.
• Use solid or dashed lines based on inequality type.
• Determine shading direction from boundary lines.