How to Solve a System of Inequalities

Question

Step 1
Graph each inequality
Step 2
Identify where the two
Answer
overlap.
1-----!
》
Step 3
Any point in that overlapping area is a
Answer
to the
system.
# solution
*half planes

Answer 1

To solve a system of inequalities, you can follow these steps in detail:

Step 1: Graph Each Inequality

• Convert each inequality to its boundary line:
  • For a linear inequality like \( ax + by < c \), first rewrite it as an equation \( ax + by = c \) to find the boundary line.
• Determine the type of line:
  • Use a dashed line for inequalities that use \( < \) or \( > \) (indicating that points on the line are not included).
  • Use a solid line for inequalities that use \( \leq \) or \( \geq \) (indicating that points on the line are included).
• Shade the correct region:
  • Choose a test point (commonly the origin \((0,0)\), if it’s not on the line) to see if it satisfies the inequality.
  • If the test point satisfies the inequality, shade the area that includes the test point; otherwise, shade the opposite side.

Step 2: Identify Where the Two Regions Overlap

• After you have graphed all inequalities, look for the region where the shaded areas overlap. This overlapping section represents the solution to the system of inequalities.

Step 3: Any Point in the Overlapping Area is a Solution

• Any point (x, y) within the overlapping shaded region is a solution that satisfies all inequalities in the system. Identify several points within this region if needed for specific solutions.

Final Explanation

• The solution to a system of inequalities can be visualized as the intersection of multiple half-planes in the coordinate graph. Each inequality divides the plane into two half-planes, and the solutions are found in the region where all half-planes intersect.

Example

Consider the system of inequalities:

1. \( y < 2x + 1 \)

2. \( y \geq -x + 2 \)

3. Graph: Graph the line \( y = 2x + 1 \) as a dashed line and shade below it.

4. Graph the line \( y = -x + 2 \) as a solid line and shade above it.

5. Overlap: Identify the region where the shading overlaps.

The points in this overlapping area are the solutions to the system of inequalities.

Answer 2

What is step 2 and 3??

Answer 3

Sure! Here's a more detailed explanation of Step 2 and Step 3 for solving a system of inequalities:

Step 2: Identify Where the Two Regions Overlap

• Look for the Intersection: After you graph all the inequalities, examine the shaded areas. The goal is to find the region where the shading overlaps for all the inequalities.
• Visualize the Overlap: The overlapping area is the solution set for the system. This region contains all the points (x, y) that satisfy all the inequalities simultaneously. You may use different colors or hatching to make it easier to see the overlapping area.
• Boundary Points: The boundary lines (whether dashed or solid) help define the edges of the solution area. Remember:
  • If the line is dashed, the points on that line are not included in the solution.
  • If the line is solid, the points on that line are included.

Step 3: Any Point in the Overlapping Area is a Solution

• Choosing Points: Any point that lies within the overlapping shaded region is a solution to the system of inequalities. You can select points within this region that satisfy all the inequalities you graphed.
• Verification: To ensure that a point is a solution, you can substitute the coordinates of the point back into each inequality. If the point satisfies all inequalities, then it is one of the solutions to the system.
• General Solution: The entire overlapping area represents infinite solutions because there are infinitely many points in that region. You can express the solution set in terms of inequalities or simply note that any point in the shaded overlap satisfies the system.

Summary

• Step 2: Identify the overlapping region on the graph that satisfies all inequalities.
• Step 3: Confirm that any point within this overlapping region is a valid solution to the system of inequalities.

This process allows you to visualize and identify the solution set for systems of inequalities effectively!

Answer 4

Answers are :
Solution
Half plane

So what is step 2 and 3

Answer 5

Thank you for clarifying! Let's restate Step 2 and Step 3 using your specified answers.

Step 2: Identify Where the Two Half Planes Overlap

• After graphing each inequality, look for the area on the graph where the shaded regions (representing half planes) from each inequality overlap. This overlapping area is crucial because it represents the region where all the conditions defined by the inequalities are met.

Step 3: Any Point in that Overlapping Area is a Solution

• Any point within the overlapping half planes satisfies all the inequalities in the system. Therefore, you can conclude that every point in this overlapping region is a solution to the system of inequalities.

Summary of Steps

• Step 2: Identify where the two half planes overlap.
• Step 3: Any point in that overlapping area is a solution to the system.

This straightforward approach emphasizes the geometric understanding of systems of inequalities as intersections of half planes.