Question

choose the system of inequalities that best matches the graph below.

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$$\begin{aligned} & y \\ge 2x + 1 \\\\ & y \\le 2x + 2 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} & y \\le \\frac{1}{2}x + 1 \\\\ & y \\ge 2x + 2 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} & y \\ge \\frac{1}{2}x + 1 \\\\ & y \\le \\frac{1}{2}x + 2 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} & y \\le 2x + 1 \\\\ & y \\ge \\frac{1}{2}x + 2 \\end{aligned}$$

\\)

Explanation:

🆕 New Concept Discovered: Graphing Systems of Linear Inequalities
Shading regions where multiple boundary lines overlap

Step 1: Find the equation of the steeper boundary line (red line)

Let's identify two points on the steeper line to find its equation:

• The \(y\)-intercept is at \((0, 1)\).
• Another clear grid intersection is at \((1, 3)\) or \((2, 5)\).

Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\):

$$m = \frac{3 - 1}{1 - 0} = 2$$

Using the slope-intercept form \(y = mx + b\):

$$y = 2x + 1$$

Now, determine the inequality direction. The shaded region is to the left/above this line. Let's test a point in the shaded region, such as \((3, 5)\):

$$5 \ge 2(3) + 1 \implies 5 \ge 7 \quad \text{(False)}$$

Let's test another point deep in the shaded region, like \((4, 6)\):

$$6 \ge 2(4) + 1 \implies 6 \ge 9 \quad \text{(False)}$$

Let's test \((3, 6)\):

$$6 \ge 2(3) + 1 \implies 6 \ge 7 \quad \text{(False)}$$

Let's look closely at the boundary: for any \(x\)-value in the shaded region (e.g., \(x = 3\)), the shaded \(y\)-values are above the flatter line but to the left of the steeper line.
For the steeper line \(y = 2x + 1\), at \(x = 2\), the line is at \(y = 5\). The shaded region at \(x = 2\) is around \(y = 3\) and \(y = 4\), which is below the line.
Let's test \((2, 3)\) which is in the shaded region:

$$3 \le 2(2) + 1 \implies 3 \le 5 \quad \text{(True)}$$

Thus, the inequality for this boundary is:

$$y \le 2x + 1$$

Step 2: Find the equation of the flatter boundary line (blue line)

Let's identify two points on the flatter line:

• The \(y\)-intercept is at \((0, 2)\).
• The \(x\)-intercept is at \((-4, 0)\).

Using these points to find the slope:

$$m = \frac{2 - 0}{0 - (-4)} = \frac{2}{4} = \frac{1}{2}$$

Using the slope-intercept form:

$$y = \frac{1}{2}x + 2$$

Now, determine the inequality direction. The shaded region is above this line. Let's test the point \((2, 3)\) which is in the shaded region:

$$3 \ge \frac{1}{2}(2) + 2 \implies 3 \ge 3 \quad \text{(True)}$$

Thus, the inequality for this boundary is:

$$y \ge \frac{1}{2}x + 2$$

Step 3: Combine the inequalities

Combining the two identified inequalities gives the system:

$$
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This matches the fourth option.

Answer:

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