Questions: Graph the solution set of the following system of inequalities. y > 5x + 10 y < x + 5

Graph the solution set of the following system of inequalities. y > 5x + 10 y < x + 5
Transcript text: Graph the solution set of the following system of inequalities. \[ \begin{array}{l} y>5 x+10 \\ y
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Solution

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Solution Steps

Step 1: Graph the first inequality
  • The first inequality is \( y \leq 5x + 10 \).
  • To graph this, start by plotting the line \( y = 5x + 10 \). This line has a slope of 5 and a y-intercept of 10.
  • Since the inequality is \( \leq \), shade the region below the line.
Step 2: Graph the second inequality
  • The second inequality is \( y < x + 5 \).
  • To graph this, start by plotting the line \( y = x + 5 \). This line has a slope of 1 and a y-intercept of 5.
  • Since the inequality is \( < \), shade the region below the line with a dashed line to indicate that the line itself is not included in the solution.
Step 3: Identify the solution region
  • 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.

Final Answer

The solution set is the region where the shaded areas from both inequalities overlap. This region can be visualized on the graph by plotting the lines and shading the appropriate areas as described in the steps above.

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