Question
Solve each set of inequalities by graphic.
1.) y less then or equal to 2x - 1
y greater then or equal to -x +3
1.) y less then or equal to 2x - 1
y greater then or equal to -x +3
Answers
GPT-4o mini
To solve the system of inequalities graphically, we will first rewrite each inequality in slope-intercept form (y = mx + b) if necessary and then sketch their graphs.
1. The first inequality is:
\[
y \leq 2x - 1
\]
This represents a line with a slope of 2 and a y-intercept of -1. The line itself will be solid because the inequality is "less than or equal to."
2. The second inequality is:
\[
y \geq -x + 3
\]
This represents a line with a slope of -1 and a y-intercept of 3. The line will also be solid because the inequality is "greater than or equal to."
### Steps to Graph:
- **Graph the first inequality \(y = 2x - 1\)**:
- Start at the y-intercept (0, -1).
- Use the slope (2) to plot another point. From (0, -1), go up 2 units and right 1 unit to (1, 1).
- Draw a solid line through these points and shade below the line to indicate that \(y\) is less than or equal to \(2x - 1\).
- **Graph the second inequality \(y = -x + 3\)**:
- Start at the y-intercept (0, 3).
- Use the slope (-1) to plot another point. From (0, 3), go down 1 unit and right 1 unit to (1, 2).
- Draw a solid line through these points and shade above the line to indicate that \(y\) is greater than or equal to \(-x + 3\).
### Find the Solution Region:
- The solution to the system of inequalities will be the region where the shaded areas of both inequalities overlap.
### Intersection Point:
To find the intersection point of the two lines, set the equations equal to each other:
\[
2x - 1 = -x + 3
\]
Solving for \(x\):
\[
2x + x = 3 + 1
\]
\[
3x = 4
\]
\[
x = \frac{4}{3}
\]
Now, substitute \(x = \frac{4}{3}\) back into either equation to find \(y\):
Using \(y = 2x - 1\):
\[
y = 2\left(\frac{4}{3}\right) - 1 = \frac{8}{3} - 1 = \frac{8}{3} - \frac{3}{3} = \frac{5}{3}
\]
So the intersection point is \(\left(\frac{4}{3}, \frac{5}{3}\right)\).
### Conclusion:
The solution region is the area where the shading overlaps. The intersection point \(\left(\frac{4}{3}, \frac{5}{3}\right)\) is located within this region. You can use graphing software or graph paper to visualize this more accurately and sketch the overlapping shaded area accordingly.
1. The first inequality is:
\[
y \leq 2x - 1
\]
This represents a line with a slope of 2 and a y-intercept of -1. The line itself will be solid because the inequality is "less than or equal to."
2. The second inequality is:
\[
y \geq -x + 3
\]
This represents a line with a slope of -1 and a y-intercept of 3. The line will also be solid because the inequality is "greater than or equal to."
### Steps to Graph:
- **Graph the first inequality \(y = 2x - 1\)**:
- Start at the y-intercept (0, -1).
- Use the slope (2) to plot another point. From (0, -1), go up 2 units and right 1 unit to (1, 1).
- Draw a solid line through these points and shade below the line to indicate that \(y\) is less than or equal to \(2x - 1\).
- **Graph the second inequality \(y = -x + 3\)**:
- Start at the y-intercept (0, 3).
- Use the slope (-1) to plot another point. From (0, 3), go down 1 unit and right 1 unit to (1, 2).
- Draw a solid line through these points and shade above the line to indicate that \(y\) is greater than or equal to \(-x + 3\).
### Find the Solution Region:
- The solution to the system of inequalities will be the region where the shaded areas of both inequalities overlap.
### Intersection Point:
To find the intersection point of the two lines, set the equations equal to each other:
\[
2x - 1 = -x + 3
\]
Solving for \(x\):
\[
2x + x = 3 + 1
\]
\[
3x = 4
\]
\[
x = \frac{4}{3}
\]
Now, substitute \(x = \frac{4}{3}\) back into either equation to find \(y\):
Using \(y = 2x - 1\):
\[
y = 2\left(\frac{4}{3}\right) - 1 = \frac{8}{3} - 1 = \frac{8}{3} - \frac{3}{3} = \frac{5}{3}
\]
So the intersection point is \(\left(\frac{4}{3}, \frac{5}{3}\right)\).
### Conclusion:
The solution region is the area where the shading overlaps. The intersection point \(\left(\frac{4}{3}, \frac{5}{3}\right)\) is located within this region. You can use graphing software or graph paper to visualize this more accurately and sketch the overlapping shaded area accordingly.