Questions: y > 2x - 2 y < -1/4 x + 2

Transcript text: $\left\{\begin{array}{l}y>2 x-2 \\ y<-\frac{1}{4} x+2\end{array}\right.$

Solution

Step 1: Graph the first inequality $ y > 2x - 2 $

Identify the boundary line: The boundary line for the inequality $ y > 2x - 2 $ is $ y = 2x - 2 $.
Plot the boundary line: Since the inequality is strict (">"), the boundary line will be dashed. Plot the line by finding two points. For example, when $ x = 0 $, $ y = -2 $ and when $ x = 1 $, $ y = 0 $.
Shade the region: Since the inequality is $ y > 2x - 2 $, shade the region above the dashed line.

Step 2: Graph the second inequality $ y < \frac{1}{4}x + 2 $

Identify the boundary line: The boundary line for the inequality $ y < \frac{1}{4}x + 2 $ is $ y = \frac{1}{4}x + 2 $.
Plot the boundary line: Since the inequality is strict ("<"), the boundary line will be dashed. Plot the line by finding two points. For example, when $ x = 0 $, $ y = 2 $ and when $ x = 4 $, $ y = 3 $.
Shade the region: Since the inequality is $ y < \frac{1}{4}x + 2 $, shade the region below the dashed line.

Step 3: Identify the solution region

Intersection of shaded regions: The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. This region represents all the points that satisfy both inequalities simultaneously.

Graph the inequalities: The final graph will show the dashed lines for both inequalities and the overlapping shaded region, which is the solution to the system of inequalities.

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