Question

multiple choice 1 point
sketch the solution to this system of inequalities:
$x - y < 3$
$2x + y < 3$
a)
image of a graph
b)
image of a graph
c)
image of a graph
d)
image of a graph

Explanation:

Step1: Analyze the first inequality \(x - y < 3\)

Rewrite it as \(y > x - 3\). The boundary line \(y = x - 3\) is dashed (since the inequality is strict, \(<\)), and we shade above the line (because \(y\) is greater than \(x - 3\)).

Step2: Analyze the second inequality \(2x + y \leq 3\)

Rewrite it as \(y \leq -2x + 3\). The boundary line \(y = -2x + 3\) is solid (since the inequality is non - strict, \(\leq\)), and we shade below the line (because \(y\) is less than or equal to \(-2x + 3\)).

Step3: Find the intersection of the two solution regions

We need to find the region that is above \(y=x - 3\) (dashed line) and below \(y=-2x + 3\) (solid line).

Now, let's analyze the options:

• Option A: The shaded region does not seem to be the intersection of the regions defined by the two inequalities.
• Option B: The shaded region is below \(y = - 2x+3\) (solid line) and above \(y=x - 3\) (dashed line), which matches the solution of the system of inequalities.
• Option C: The shaded region does not match the intersection of the two solution regions.
• Option D: The shaded region does not match the intersection of the two solution regions.

Answer:

B. (The option corresponding to the graph where the shaded region is below \(y = - 2x + 3\) (solid line) and above \(y=x - 3\) (dashed line))