Graph the solution set of the system of inequalities.
3 x minus 4 y less than or equals 12
5 x minus 2 y greater than 10

To graph the solution set of the system of inequalities:

### Inequalities:
1. \( 3x - 4y \leq 12 \)
2. \( 5x - 2y > 10 \)

### Step-by-step process:

#### 1. Rearrange both inequalities into slope-intercept form (i.e., \( y = mx + b \)):

##### First inequality: \( 3x - 4y \leq 12 \)
- Subtract \( 3x \) from both sides:  
  \[ -4y \leq -3x + 12 \]
- Divide both sides by \( -4 \) (note that dividing by a negative flips the inequality):  
  \[ y \geq \frac{3}{4}x - 3 \]

##### Second inequality: \( 5x - 2y > 10 \)
- Subtract \( 5x \) from both sides:  
  \[ -2y > -5x + 10 \]
- Divide both sides by \( -2 \) (again, the inequality flips):  
  \[ y < \frac{5}{2}x - 5 \]

Now, we have:
- \( y \geq \frac{3}{4}x - 3 \) 
- \( y < \frac{5}{2}x - 5 \)

#### 2. Graph the boundary lines:
- \( y = \frac{3}{4}x - 3 \) (solid line since it is \( \leq \))
- \( y = \frac{5}{2}x - 5 \) (dashed line since it is \( > \))

#### 3. Shade the solution regions:
- For \( y \geq \frac{3}{4}x - 3 \), shade above the line.
- For \( y < \frac{5}{2}x - 5 \), shade below the line.

The solution set is where the shaded regions overlap.

Let me now graph it.It seems like I can’t do more advanced data analysis right now. Please try again later. If you have any further questions, feel free to ask!

Here are some related questions to expand on this topic:
1. How do you determine which region to shade in a graph of inequalities?
2. What does it mean when the line is solid versus dashed in inequality graphs?
3. How do you solve systems of inequalities algebraically without graphing?
4. Can systems of inequalities have no solution? How would that appear on a graph?
5. How can inequalities be applied in real-world scenarios?

**Tip:** When graphing inequalities, always remember to check whether the boundary line should be solid (≤ or ≥) or dashed (< or >).

Graph the solution set of the system of inequalities. 3x - 4y ≤ 12, 5x - 2y > 10

Learn how to graph the solution set of the system of inequalities 3x - 4y ≤ 12 and 5x - 2y > 10. Follow a step-by-step breakdown of converting inequalities into slope-intercept form, graphing boundary lines, and shading solution regions. This problem is ideal for students in Grades 9-11 covering algebraic graphing concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation