Question

related questions: lesson 5: inequalities

click on the area that would be within the solution set for this system of equations.

q. \\(\

$$\begin{cases} y < 2x + 1 \\\\ y \\ge -x - 2 \\end{cases}$$

\\)

Explanation:

Identify the boundary lines

Using the Systems of Linear Inequalities knowledge point, we identify the two boundary lines from the given system:

1. The first boundary line is \(y = 2x + 1\). It has a \(y\)-intercept at \((0, 1)\) and a slope of \(2\). It is represented by the dashed blue line.
2. The second boundary line is \(y = -x - 2\). It has a \(y\)-intercept at \((0, -2)\) and a slope of \(-1\). It is represented by the solid blue line.

Analyze the first inequality

Using the Systems of Linear Inequalities knowledge point, we analyze the region for:

$$y < 2x + 1$$

Since the inequality symbol is \(<\), the solution region lies strictly below the dashed boundary line \(y = 2x + 1\).

Analyze the second inequality

Using the Systems of Linear Inequalities knowledge point, we analyze the region for:

$$y \ge -x - 2$$

Since the inequality symbol is \(\ge\), the solution region lies on or above the solid boundary line \(y = -x - 2\).

Find the overlapping feasible region

Using the Feasible Region Identification knowledge point, we find the intersection of the two regions:

• Below the dashed line \(y = 2x + 1\)
• Above the solid line \(y = -x - 2\)

Let's test a point in this overlapping region, for example, \((2, 0)\):

• First inequality: \(0 < 2(2) + 1 \implies 0 < 5\) (True)
• Second inequality: \(0 \ge -2 - 2 \implies 0 \ge -4\) (True)

Thus, the region containing the point \((2, 0)\) (the right-hand wedge between the two lines) is the correct solution set.

Answer:

The solution set is the region that lies below the dashed line \(y = 2x + 1\) and above the solid line \(y = -x - 2\).

On the graph, this corresponds to the right-hand region containing points such as \((2, 0)\), \((3, -1)\), and \((1, -1)\).