Math Problem Statement

Use the given graphs of y1 and y2 to solve each inequality. Write the solution set in interval notation.

Solution

To solve the inequalities using the given graphs of y1y_1 and y2y_2, let's analyze the intersection and relative positions of the two lines:

  • The blue line represents y1y_1 and the red line represents y2y_2.
  • The lines intersect at the point (2,2)(-2, -2).

Now, we will solve the three parts based on this information:

(a) y1=y2y_1 = y_2

This happens at the point where the two lines intersect, which is at (2,2)(-2, -2).

Solution: {2}\{ -2 \} in set notation or [2,2][-2, -2] in interval notation.

(b) y1>y2y_1 > y_2

The blue line (y1y_1) is greater than the red line (y2y_2) to the left of the intersection, i.e., when x<2x < -2.

Solution: (,2)(-\infty, -2)

(c) y1y2y_1 \leq y_2

This inequality holds where y1y_1 is less than or equal to y2y_2, which occurs when x2x \geq -2.

Solution: [2,)[-2, \infty)

Would you like more details on any of the steps? Here are 5 related questions:

  1. How do we find the intersection of two lines algebraically?
  2. What methods can be used to solve linear inequalities graphically?
  3. How would the solution change if the slope of one of the lines was different?
  4. What is the significance of the interval notation used in inequalities?
  5. Can you always determine inequalities between lines by looking at their graphs?

Tip: When solving inequalities graphically, always pay attention to the intersection points, as they often mark transitions in the inequality signs.

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Graphing
Coordinate Geometry

Formulas

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Theorems

Intersection Theorem
Graphical Solution of Inequalities

Suitable Grade Level

Grades 9-12