Question
Illustrate the following on graph paper and shade the region which satisfies all the three inequalities at the same time: -x + 5y is less than or equal to 10, 3x - 4y is less than or equal to 8 and x > -1
Answers
GPT-4o mini
To illustrate the inequalities and shade the region that satisfies all three conditions, follow these steps:
### Step 1: Rewrite the Inequalities
1. **First Inequality**: \(-x + 5y \leq 10\)
- Rearranging gives: \(5y \leq x + 10\) → \(y \leq \frac{1}{5}x + 2\)
2. **Second Inequality**: \(3x - 4y \leq 8\)
- Rearranging gives: \(-4y \leq -3x + 8\) → \(y \geq \frac{3}{4}x - 2\)
3. **Third Inequality**: \(x > -1\)
### Step 2: Graph Each Inequality
1. **Graph \(y \leq \frac{1}{5}x + 2\)**:
- This is a line with a slope of \(\frac{1}{5}\) and a y-intercept of 2. Since it's “less than or equal to,” the area below the line should be shaded.
2. **Graph \(y \geq \frac{3}{4}x - 2\)**:
- This is a line with a slope of \(\frac{3}{4}\) and a y-intercept of -2. Since it's “greater than or equal to,” shade the area above the line.
3. **Graph \(x > -1\)**:
- This represents a vertical line at \(x = -1\). Since it's strict inequality (greater than), shade the region to the right of this line but do not include the line itself.
### Step 3: Find the Intersection of the Regions
- The solution to the system of inequalities will be where the shaded regions for all three inequalities overlap.
### Step 4: Shade the Feasible Region
- Look for the area that is below the line \(y = \frac{1}{5}x + 2\), above the line \(y = \frac{3}{4}x - 2\), and to the right of the line \(x = -1\). This intersection is the feasible region that satisfies all three inequalities simultaneously.
### Important Note
Since I cannot illustrate directly on graph paper, here’s how you can do so visually:
1. Draw the coordinate axes.
2. Plot the line for the first inequality and shade below it.
3. Plot the line for the second inequality and shade above it.
4. Draw a dashed vertical line at \(x = -1\) and shade to the right of it.
Make sure to label the lines and shaded areas accurately. The region where all the shading overlaps represents the solution to the system of inequalities.
### Step 1: Rewrite the Inequalities
1. **First Inequality**: \(-x + 5y \leq 10\)
- Rearranging gives: \(5y \leq x + 10\) → \(y \leq \frac{1}{5}x + 2\)
2. **Second Inequality**: \(3x - 4y \leq 8\)
- Rearranging gives: \(-4y \leq -3x + 8\) → \(y \geq \frac{3}{4}x - 2\)
3. **Third Inequality**: \(x > -1\)
### Step 2: Graph Each Inequality
1. **Graph \(y \leq \frac{1}{5}x + 2\)**:
- This is a line with a slope of \(\frac{1}{5}\) and a y-intercept of 2. Since it's “less than or equal to,” the area below the line should be shaded.
2. **Graph \(y \geq \frac{3}{4}x - 2\)**:
- This is a line with a slope of \(\frac{3}{4}\) and a y-intercept of -2. Since it's “greater than or equal to,” shade the area above the line.
3. **Graph \(x > -1\)**:
- This represents a vertical line at \(x = -1\). Since it's strict inequality (greater than), shade the region to the right of this line but do not include the line itself.
### Step 3: Find the Intersection of the Regions
- The solution to the system of inequalities will be where the shaded regions for all three inequalities overlap.
### Step 4: Shade the Feasible Region
- Look for the area that is below the line \(y = \frac{1}{5}x + 2\), above the line \(y = \frac{3}{4}x - 2\), and to the right of the line \(x = -1\). This intersection is the feasible region that satisfies all three inequalities simultaneously.
### Important Note
Since I cannot illustrate directly on graph paper, here’s how you can do so visually:
1. Draw the coordinate axes.
2. Plot the line for the first inequality and shade below it.
3. Plot the line for the second inequality and shade above it.
4. Draw a dashed vertical line at \(x = -1\) and shade to the right of it.
Make sure to label the lines and shaded areas accurately. The region where all the shading overlaps represents the solution to the system of inequalities.