Questions: Find the coordinate plane that represents the solution of this system. y > -x + 5 y + 1 >= -x

Find the coordinate plane that represents the solution of this system. y > -x + 5 y + 1 >= -x
Transcript text: Find the coordinate plane that represents the solution of this system. \[ \begin{array}{l} y>-x+5 \\ y+1 \geq-x \end{array} \]
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Solution

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Solution Steps

To solve the system of inequalities, we need to determine the region on the coordinate plane that satisfies both inequalities. This involves graphing each inequality and finding the intersection of the regions they define. The first inequality, \( y > -x + 5 \), represents a region above the line \( y = -x + 5 \). The second inequality, \( y + 1 \geq -x \), simplifies to \( y \geq -x - 1 \), representing a region above or on the line \( y = -x - 1 \). The solution is the area where these two regions overlap.

Step 1: Analyze the Inequalities

We are given the system of inequalities:

  1. \( y > -x + 5 \)
  2. \( y + 1 \geq -x \)

Our goal is to find the region on the coordinate plane that satisfies both inequalities.

Step 2: Rewrite the Inequalities

First, let's rewrite the second inequality in a more standard form:

\[ y + 1 \geq -x \implies y \geq -x - 1 \]

Now, we have the system:

  1. \( y > -x + 5 \)
  2. \( y \geq -x - 1 \)
Step 3: Graph the Boundary Lines

For each inequality, we will graph the boundary line and determine which side of the line satisfies the inequality.

Inequality 1: \( y > -x + 5 \)
  • The boundary line is \( y = -x + 5 \).
  • This is a line with a slope of \(-1\) and a y-intercept of \(5\).
  • Since the inequality is strict (\(>\)), we will use a dashed line to represent the boundary.
Inequality 2: \( y \geq -x - 1 \)
  • The boundary line is \( y = -x - 1 \).
  • This is a line with a slope of \(-1\) and a y-intercept of \(-1\).
  • Since the inequality is non-strict (\(\geq\)), we will use a solid line to represent the boundary.
Step 4: Determine the Solution Region

For each inequality, determine which side of the boundary line satisfies the inequality:

  • For \( y > -x + 5 \), the region above the line \( y = -x + 5 \) satisfies the inequality.
  • For \( y \geq -x - 1 \), the region above or on the line \( y = -x - 1 \) satisfies the inequality.
Step 5: Find the Intersection of the Regions

The solution to the system of inequalities is the region where the solutions to both inequalities overlap.

  • The region above the line \( y = -x + 5 \) and above or on the line \( y = -x - 1 \) is the solution region.

Final Answer

The solution region is the area above the line \( y = -x + 5 \) and above or on the line \( y = -x - 1 \). This region can be represented on the coordinate plane as the area that satisfies both inequalities simultaneously.

\[ \boxed{\text{The solution is the region above } y = -x + 5 \text{ and above or on } y = -x -1.} \]

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