Question
Solve each system of inequalities by graphing.
3x + 4y less then -14
x - 3y greater then or equal to 17
3x + 4y less then -14
x - 3y greater then or equal to 17
Answers
GPT-4o mini
To solve the system of inequalities by graphing, follow these steps:
1. **Rewrite the inequalities**:
We will rewrite the inequalities in slope-intercept form (y = mx + b), which makes them easier to graph.
**First inequality:**
\(3x + 4y < -14\)
Solving for y:
\[
4y < -3x - 14
\]
\[
y < -\frac{3}{4}x - \frac{14}{4}
\]
\[
y < -\frac{3}{4}x - \frac{7}{2}
\]
**Second inequality:**
\(x - 3y \geq 17\)
Solving for y:
\[
-3y \geq -x + 17
\]
\[
y \leq \frac{1}{3}x - \frac{17}{3}
\]
2. **Graph the inequalities**:
**Inequality 1: \(y < -\frac{3}{4}x - \frac{7}{2}\)**
- The boundary line is \(y = -\frac{3}{4}x - \frac{7}{2}\). This line will be dashed because the inequality is strict ("less than").
- Find two points to graph the line. For example:
- If \(x = 0\), then \(y = -\frac{7}{2} = -3.5\).
- If \(x = 4\), then \(y = -\frac{3}{4}(4) - \frac{7}{2} = -3 - 3.5 = -6.5\).
- Plot these points (0, -3.5) and (4, -6.5), and draw a dashed line through them. Shade the region below this line.
**Inequality 2: \(y \leq \frac{1}{3}x - \frac{17}{3}\)**
- The boundary line is \(y = \frac{1}{3}x - \frac{17}{3}\). This line will be solid because the inequality is inclusive ("greater than or equal to").
- Find two points to graph the line. For example:
- If \(x = 0\), then \(y = -\frac{17}{3} \approx -5.67\).
- If \(x = 6\), then \(y = \frac{1}{3}(6) - \frac{17}{3} = 2 - \frac{17}{3} = 2 - 5.67 = -3.67\).
- Plot these points (0, -5.67) and (6, -3.67), and draw a solid line through them. Shade the region below this line.
3. **Identify the solution region**:
- The solution to the system of inequalities is the intersection of the shaded areas from both inequalities. This is the region where both conditions are met.
### Summary
- The first inequality's region will be below the dashed line \(y = -\frac{3}{4}x - \frac{7}{2}\).
- The second inequality's region will be below the solid line \(y = \frac{1}{3}x - \frac{17}{3}\).
Make sure to check a few points in the overlapping shaded region to confirm they satisfy both inequalities.
1. **Rewrite the inequalities**:
We will rewrite the inequalities in slope-intercept form (y = mx + b), which makes them easier to graph.
**First inequality:**
\(3x + 4y < -14\)
Solving for y:
\[
4y < -3x - 14
\]
\[
y < -\frac{3}{4}x - \frac{14}{4}
\]
\[
y < -\frac{3}{4}x - \frac{7}{2}
\]
**Second inequality:**
\(x - 3y \geq 17\)
Solving for y:
\[
-3y \geq -x + 17
\]
\[
y \leq \frac{1}{3}x - \frac{17}{3}
\]
2. **Graph the inequalities**:
**Inequality 1: \(y < -\frac{3}{4}x - \frac{7}{2}\)**
- The boundary line is \(y = -\frac{3}{4}x - \frac{7}{2}\). This line will be dashed because the inequality is strict ("less than").
- Find two points to graph the line. For example:
- If \(x = 0\), then \(y = -\frac{7}{2} = -3.5\).
- If \(x = 4\), then \(y = -\frac{3}{4}(4) - \frac{7}{2} = -3 - 3.5 = -6.5\).
- Plot these points (0, -3.5) and (4, -6.5), and draw a dashed line through them. Shade the region below this line.
**Inequality 2: \(y \leq \frac{1}{3}x - \frac{17}{3}\)**
- The boundary line is \(y = \frac{1}{3}x - \frac{17}{3}\). This line will be solid because the inequality is inclusive ("greater than or equal to").
- Find two points to graph the line. For example:
- If \(x = 0\), then \(y = -\frac{17}{3} \approx -5.67\).
- If \(x = 6\), then \(y = \frac{1}{3}(6) - \frac{17}{3} = 2 - \frac{17}{3} = 2 - 5.67 = -3.67\).
- Plot these points (0, -5.67) and (6, -3.67), and draw a solid line through them. Shade the region below this line.
3. **Identify the solution region**:
- The solution to the system of inequalities is the intersection of the shaded areas from both inequalities. This is the region where both conditions are met.
### Summary
- The first inequality's region will be below the dashed line \(y = -\frac{3}{4}x - \frac{7}{2}\).
- The second inequality's region will be below the solid line \(y = \frac{1}{3}x - \frac{17}{3}\).
Make sure to check a few points in the overlapping shaded region to confirm they satisfy both inequalities.