Question

the following system of linear inequalities is graphed below. describe the solution in words.
-4x + 2y ≥ 6
2x - y > 2
describe the solution to the system of inequalities whose graph is shown at left.
a. the solution is 2x - y > 2.
b. the solution is -4x + 2y ≥ 6.
c. the solution is all real numbers.
d. there is no solution.

Explanation:

Step1: Analyze the system of inequalities

The system is \(

$$\begin{cases}-4x + 2y\geq6\\2x - y>2\end{cases}$$

\). Let's rewrite the inequalities in slope - intercept form (\(y = mx + b\)).

For \(-4x + 2y\geq6\), add \(4x\) to both sides: \(2y\geq4x + 6\), then divide by 2: \(y\geq2x+3\).

For \(2x - y>2\), subtract \(2x\) from both sides: \(-y>-2x + 2\), multiply both sides by - 1 (and reverse the inequality sign): \(y<2x - 2\).

Step2: Analyze the regions of the inequalities

The inequality \(y\geq2x + 3\) represents the region above (including the line) the line \(y = 2x+3\). The inequality \(y<2x - 2\) represents the region below the line \(y=2x - 2\).

The line \(y = 2x+3\) has a slope of 2 and a y - intercept of 3. The line \(y=2x - 2\) has a slope of 2 and a y - intercept of - 2. Since the two lines are parallel (same slope) and the region above \(y = 2x + 3\) and the region below \(y=2x - 2\) do not overlap (because \(2x+3>2x - 2\) for all real \(x\)), there is no point \((x,y)\) that satisfies both inequalities simultaneously.

Answer:

D. There is no solution.