Math Problem Statement

Which of the following inequalities is graphed on the coordinate plane?

Solution

To determine which inequality corresponds to the shaded region in the graph, let's analyze the line and shading shown in the coordinate plane.

Step 1: Identify the Line Equation

From the graph, the line appears to have:

  • A y-intercept at 66 (point (0,6)(0,6)).
  • A slope that goes downwards by 2 units for every 1 unit it moves right, giving a slope of 2-2.

This means the line equation is: y=2x+6y = -2x + 6

Step 2: Determine the Inequality

  • The shading is on the region below the line, which suggests a “less than” inequality.
  • Therefore, the inequality describing this shaded area would be: y<2x+6y < -2x + 6

Step 3: Match with the Options

Looking at the provided options:

  • Option A: y2x+6y \leq -2x + 6 (matches our inequality, but it should be strict inequality without “equal to”)
  • Option C: y<2x+6y < 2x + 6 (incorrect slope sign)
  • Option D: y>2x+6y > 2x + 6 (incorrect slope and inequality direction)

Answer:

The correct answer that fits the graph’s shaded region and line is: Option A: y2x+6\text{Option A: } y \leq -2x + 6

Would you like further details or have any questions?

Here are some related questions that expand on this concept:

  1. How would the inequality change if the shaded region were above the line?
  2. How does changing the slope affect the inequality?
  3. What does a dashed line mean in an inequality graph?
  4. How do you determine the correct inequality sign based on shading?
  5. What role does the y-intercept play in graphing linear inequalities?

Tip: Always check if the line is solid or dashed; a solid line includes points on the line (≤ or ≥), while a dashed line excludes them (< or >).

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Math Problem Analysis

Mathematical Concepts

Algebra
Graphing Linear Inequalities

Formulas

Slope-intercept form y = mx + b

Theorems

Linear inequality graphing

Suitable Grade Level

Grades 8-10