Question

find the graph of this system of linear inequalities. \\(\

$$\begin{cases} y > -3x + 2 \\\\ y \\geq -1 \\end{cases}$$

\\)

Explanation:

Step1: Analyze \( y > -3x + 2 \)

The inequality \( y > -3x + 2 \) is a linear inequality. The boundary line is \( y=-3x + 2 \), which has a slope of \(-3\) and a y - intercept of \( 2 \). Since the inequality is \( y > -3x+2 \) (strict inequality), the boundary line should be dashed. To determine the region, we can test a point, for example, the origin \((0,0)\). Substitute \( x = 0,y = 0\) into \( y > -3x + 2 \): \( 0>-3(0)+2\) which is \( 0 > 2\), false. So we shade the region that does not contain the origin.

Step2: Analyze \( y\geq - 1 \)

The inequality \( y\geq - 1 \) has a boundary line \( y=-1 \), which is a horizontal line. Since the inequality is \( y\geq - 1 \) (non - strict inequality), the boundary line is solid. The region that satisfies this inequality is the area above (including the line) \( y=-1 \).

Step3: Find the intersection of the two regions

We need to find the region that satisfies both inequalities. The region for \( y > -3x + 2 \) (dashed line, region not containing origin) and the region for \( y\geq - 1 \) (solid line, region above \( y = - 1\)) should overlap.

Looking at the three graphs:

• For the first graph: The dashed line is \( y=-3x + 2 \), the solid line is \( y=-1 \). The region that is above \( y=-1 \) (since \( y\geq - 1\)) and above the dashed line \( y=-3x + 2 \) (since \( y > -3x + 2\)) is the purple region. Let's check the shading: The region for \( y\geq - 1\) is above \( y=-1\), and the region for \( y > -3x + 2\) (dashed line) when combined with \( y\geq - 1\) gives the purple area in the first graph.
• For the second graph: The shading for \( y > -3x + 2\) is incorrect (it seems to contain the origin - like region) and the combination with \( y\geq - 1\) is wrong.
• For the third graph: The shading for \( y > -3x + 2\) is incorrect (it contains the origin - like region) and the combination with \( y\geq - 1\) is wrong.

Answer:

The first graph (the one with the purple region above \( y = - 1\) and above the dashed line \( y=-3x + 2\))