Question
Solve each system of inequalities by graphing.
x - 5y - 6 greater then or equal to 0
2x + 4y + 1 less then or equal to -1
x - 5y - 6 greater then or equal to 0
2x + 4y + 1 less then or equal to -1
Answers
GPT-4o mini
To solve the given system of inequalities by graphing, let's analyze each inequality separately and then plot them on a coordinate plane.
### Inequality 1:
**x - 5y - 6 ≥ 0**
1. Rearranging the inequality:
\[
x - 6 ≥ 5y
\]
or equivalently,
\[
5y ≤ x - 6
\]
or
\[
y ≤ \frac{1}{5}x - \frac{6}{5}
\]
2. To graph this inequality, first plot the line \(y = \frac{1}{5}x - \frac{6}{5}\). This line has a y-intercept of \(-\frac{6}{5}\) (or -1.2) and a slope of \(\frac{1}{5}\).
3. Since it's a "greater than or equal to" inequality, we will draw a solid line.
4. The region that satisfies the inequality \(y ≤ \frac{1}{5}x - \frac{6}{5}\) will be below the line.
### Inequality 2:
**2x + 4y + 1 ≤ -1**
1. Rearranging the inequality:
\[
2x + 4y ≤ -2
\]
or equivalently,
\[
4y ≤ -2 - 2x
\]
or
\[
y ≤ -\frac{1}{2}x - \frac{1}{2}
\]
2. To graph this inequality, first plot the line \(y = -\frac{1}{2}x - \frac{1}{2}\). This line has a y-intercept of \(-\frac{1}{2}\) and a slope of \(-\frac{1}{2}\).
3. Since it's a "less than or equal to" inequality, we will also draw a solid line.
4. The region that satisfies the inequality \(y ≤ -\frac{1}{2}x - \frac{1}{2}\) will be below the line.
### Graphing Together
To summarize, you'll plot these lines:
1. \(y = \frac{1}{5}x - \frac{6}{5}\) — solid line, shaded below.
2. \(y = -\frac{1}{2}x - \frac{1}{2}\) — solid line, shaded below.
### Intersection of Shaded Regions
The solution to the system of inequalities is where the shaded regions (the solutions of both inequalities) overlap.
### Conclusion
Graph these two lines on a coordinate plane and shade the appropriate regions. The area where the shadings from the two inequalities intersect will be the solution to the system of inequalities.
Note: For an exact solution, you would plot it on graphing software or graph paper to visualize the intersection region accurately.
### Inequality 1:
**x - 5y - 6 ≥ 0**
1. Rearranging the inequality:
\[
x - 6 ≥ 5y
\]
or equivalently,
\[
5y ≤ x - 6
\]
or
\[
y ≤ \frac{1}{5}x - \frac{6}{5}
\]
2. To graph this inequality, first plot the line \(y = \frac{1}{5}x - \frac{6}{5}\). This line has a y-intercept of \(-\frac{6}{5}\) (or -1.2) and a slope of \(\frac{1}{5}\).
3. Since it's a "greater than or equal to" inequality, we will draw a solid line.
4. The region that satisfies the inequality \(y ≤ \frac{1}{5}x - \frac{6}{5}\) will be below the line.
### Inequality 2:
**2x + 4y + 1 ≤ -1**
1. Rearranging the inequality:
\[
2x + 4y ≤ -2
\]
or equivalently,
\[
4y ≤ -2 - 2x
\]
or
\[
y ≤ -\frac{1}{2}x - \frac{1}{2}
\]
2. To graph this inequality, first plot the line \(y = -\frac{1}{2}x - \frac{1}{2}\). This line has a y-intercept of \(-\frac{1}{2}\) and a slope of \(-\frac{1}{2}\).
3. Since it's a "less than or equal to" inequality, we will also draw a solid line.
4. The region that satisfies the inequality \(y ≤ -\frac{1}{2}x - \frac{1}{2}\) will be below the line.
### Graphing Together
To summarize, you'll plot these lines:
1. \(y = \frac{1}{5}x - \frac{6}{5}\) — solid line, shaded below.
2. \(y = -\frac{1}{2}x - \frac{1}{2}\) — solid line, shaded below.
### Intersection of Shaded Regions
The solution to the system of inequalities is where the shaded regions (the solutions of both inequalities) overlap.
### Conclusion
Graph these two lines on a coordinate plane and shade the appropriate regions. The area where the shadings from the two inequalities intersect will be the solution to the system of inequalities.
Note: For an exact solution, you would plot it on graphing software or graph paper to visualize the intersection region accurately.