Math Problem Statement

Which is the graph of the linear inequality 2x – 3y < 12?

On a coordinate plane, a solid straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the right of the line is shaded.

On a coordinate plane, a dashed straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the right of the line is shaded.

On a coordinate plane, a dashed straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the left of the line is shaded.

On a coordinate plane, a solid straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the left of the line is shaded.

Solution

To determine the correct graph for the inequality 2x3y<122x - 3y < 12, we can first rewrite the inequality in slope-intercept form, i.e., y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Step 1: Rewrite the inequality

Start by solving for yy: 2x3y<122x - 3y < 12 Subtract 2x2x from both sides: 3y<2x+12-3y < -2x + 12 Now, divide by 3-3 (remember that dividing by a negative number flips the inequality sign): y>23x4y > \frac{2}{3}x - 4 So, the inequality becomes: y>23x4y > \frac{2}{3}x - 4

Step 2: Analyze the components

  1. Slope and intercept: The slope is 23\frac{2}{3} (positive slope), and the y-intercept is 4-4 (the line passes through the point (0,4)(0, -4)).
  2. Dashed line: Since the inequality is a strict inequality (>>, not \geq), the line is dashed, not solid.
  3. Shaded region: The inequality says y>23x4y > \frac{2}{3}x - 4, which means we shade the region above the line. This corresponds to everything to the left of the line because of how the line is oriented with a positive slope.

Step 3: Choose the correct graph

The correct graph should have:

  • A dashed line with a positive slope, going through (0,4)(0, -4) and (3,2)(3, -2).
  • The shading on the left side of the line (above the line for y>y >).

Therefore, the correct option is: "On a coordinate plane, a dashed straight line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything to the left of the line is shaded."

Do you want more details or have any questions?

Here are some related questions:

  1. How would the graph change if the inequality was 2x3y122x - 3y \geq 12?
  2. What happens if the inequality was flipped to 2x3y>122x - 3y > 12?
  3. How do you graph an inequality with a negative slope?
  4. What is the difference between shading above or below the line for an inequality?
  5. How can you determine which side of the line to shade for any inequality?

Tip: Always flip the inequality sign when dividing or multiplying by a negative number during your solving process.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Graphing on Coordinate Plane
Slope-Intercept Form

Formulas

Slope-intercept form: y = mx + b
Solving linear inequalities

Theorems

Inequality rules: flipping the sign when dividing or multiplying by a negative

Suitable Grade Level

Grades 8-10