Question

select the correct answer.

which system of inequalities has this graph as its solution?

\\(\

$$\begin{aligned} y &< 2x - 3 \\\\ y &< \\frac{1}{3}x + 4 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} y &> 2x - 3 \\\\ y &> \\frac{1}{3}x + 4 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} y &> 2x - 3 \\\\ y &< \\frac{1}{3}x + 4 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} y &< 2x - 3 \\\\ y &> \\frac{1}{3}x + 4 \\end{aligned}$$

\\)

Explanation:

Identify the boundary lines

We analyze the two dashed boundary lines shown on the coordinate plane using Slope-Intercept Form.

The first boundary line has a \(y\)-intercept at \((0, -3)\) and passes through \((3, 3)\).
The slope \(m_1\) is:

$$m_1 = \frac{3 - (-3)}{3 - 0} = \frac{6}{3} = 2$$

So, the equation of this boundary line is:

$$y = 2x - 3$$

The second boundary line has a \(y\)-intercept at \((0, 4)\) and passes through \((3, 5)\).
The slope \(m_2\) is:

$$m_2 = \frac{5 - 4}{3 - 0} = \frac{1}{3}$$

So, the equation of this boundary line is:

$$y = \frac{1}{3}x + 4$$

Both boundary lines are dashed, meaning the inequalities use strict inequalities (\(<\) or \(>\)).

Determine the inequality directions

We use a test point in the shaded region to determine the correct inequality signs for Graphing Linear Inequalities.

The point \((-5, 5)\) lies clearly within the shaded region. Let's test this point in both boundary equations:

For the line \(y = 2x - 3\):

$$5 \text{ vs } 2(-5) - 3$$

$$5 > -13$$

Since \(5 > -13\), the inequality is:

$$y > 2x - 3$$

For the line \(y = \frac{1}{3}x + 4\):

$$5 \text{ vs } \frac{1}{3}(-5) + 4$$

$$5 \text{ vs } -\frac{5}{3} + \frac{12}{3}$$

$$5 > \frac{7}{3}$$

Since \(5 > \frac{7}{3}\), the inequality is:

$$y > \frac{1}{3}x + 4$$

Match with the given options

The system of inequalities representing the shaded region is:

$$
LATEXBLOCK0
$$

This matches the second option.

Answer:

• (A) \(

$$\begin{aligned} y &< 2x - 3 \\ y &< \frac{1}{3}x + 4 \end{aligned}$$

\)

• **(B) \(

$$\begin{aligned} y &> 2x - 3 \\ y &> \frac{1}{3}x + 4 \end{aligned}$$

\) (Correct answer)**

• (C) \(

$$\begin{aligned} y &> 2x - 3 \\ y &< \frac{1}{3}x + 4 \end{aligned}$$

\)

• (D) \(

$$\begin{aligned} y &< 2x - 3 \\ y &> \frac{1}{3}x + 4 \end{aligned}$$

\)