Visually completing solutions of systems of two variables

TL;DR

Determine the x or y values that satisfy the system of inequalities represented by the graph.

Transcript

• [Voiceover] Which x-values make the ordered pair, x comma negative two, a solution of the system of inequalities represented by the graph below? So let's look at this. So we're constraining ourselves to all of the points of the form x comma negative two which is another way of saying we're going to constrain ourselves to y equaling negative two. ... Read More

Key Insights

• 💚 The graph represents two systems of inequalities: green and blue.
• 😥 Points in the shaded area of each system are solutions to the respective inequalities.
• 🫥 Points on dashed lines are not part of the solution set.
• 😥 Points that satisfy both inequalities are solutions to the system.
• ❣️ The x or y values that satisfy the system depend on which variable is constrained in the problem.
• ❣️ In the first question, y is constrained to -2, so x must be greater than or equal to -3 to be a solution.

Questions & Answers

Q: What does it mean for a point to be a solution to a system of inequalities represented by a graph?

A point is a solution if it lies within the shaded area for each inequality in the system. Points on dashed lines or outside the shaded areas are not solutions.

Q: Why does the point on the solid blue line satisfy both inequalities?

The solid blue line is included in the solution set, so any point on that line satisfies the blue inequality. Since the point is also within the shaded area for the green inequality, it satisfies both.

Q: How do we determine the x-values that make y equal to -2 a solution to the system?

We need to find all the x-values that are within or to the right of the line where y equals -2. In this case, x must be greater than or equal to -3.

Q: What are the constraints on y when x is equal to 4 in order to satisfy the system of inequalities?

For x to be equal to 4, y must be greater than -1 and less than or equal to 3. Any y-value within that range would make the ordered pair (4, y) a solution.

Summary & Key Takeaways

• The graph represents two systems of inequalities: the green system and the blue system.

• Points in the shaded area of each system are solutions to the respective inequalities.

• Points that satisfy both the green and blue inequalities are part of the solution set for the system.