Math Problem Statement

Which of the following inequalities is graphed in the xy-plane above?

Solution

Let's analyze the graph and the provided options to determine which inequality corresponds to the shaded region.

Observations:

  1. The shaded region is above the line y=2x3y = 2x - 3 (this line passes through points like (0,3)(0, -3) and (2,1)(2, 1)).
  2. The shading suggests that the inequality involves y2x3y \geq 2x - 3 because the shading is above the line.
  3. The shaded region is also constrained horizontally at x2x \geq 2. The boundary line x=2x = 2 is a vertical line, and the shading is to the right of it.

Solution:

The inequality that describes this graph is: x2andy2x3x \geq 2 \quad \text{and} \quad y \geq 2x - 3 This matches the third option: x2,y2x3x \geq 2, \, y \geq 2x - 3.


Final Answer:

The correct choice is x2,y2x3x \geq 2, \, y \geq 2x - 3.


Would you like me to explain how inequalities are interpreted in graphs or clarify any specific part?

5 Related Questions:

  1. How do you determine if shading corresponds to yy \geq or yy \leq in a graph?
  2. What does the vertical line x=2x = 2 represent in terms of inequalities?
  3. How do you find the equation of a line given two points?
  4. What happens when two inequalities overlap in a graph?
  5. How can you test if a point satisfies a given inequality?

Tip:

To confirm a solution, always test a point in the shaded region to see if it satisfies the inequality!

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Math Problem Analysis

Mathematical Concepts

Graphing Inequalities
Linear Equations
Coordinate Geometry

Formulas

Equation of a line: y = mx + b
Inequality forms: y ≥ mx + b or y ≤ mx + b

Theorems

Slope-Intercept Form
Inequalities in Coordinate Geometry

Suitable Grade Level

Grades 8-10