Question
Solve each set of inequalities by graphing.
-y less then or equal to 3x +4
-3x +3y less then or equal to 9
-y less then or equal to 3x +4
-3x +3y less then or equal to 9
Answers
GPT-4o mini
To solve the set of inequalities by graphing, we will first rewrite them for clarity and then graph each individual inequality.
1. The first inequality:
\[
-y \leq 3x + 4
\]
We can rewrite it as:
\[
y \geq -3x - 4
\]
2. The second inequality:
\[
-3x + 3y \leq 9
\]
We can simplify this by dividing the entire inequality by 3:
\[
-x + y \leq 3
\]
or
\[
y \leq x + 3
\]
Now we have the two inequalities to graph:
1. \( y \geq -3x - 4 \)
2. \( y \leq x + 3 \)
### Step 1: Graphing the first inequality
1. **Graph \( y = -3x - 4 \)**:
- The y-intercept is -4 (point (0, -4)).
- The slope is -3, which means for each increase of 1 in \( x \), \( y \) decreases by 3.
Plot the y-intercept and use the slope to find another point:
- When \( x = 1 \):
\[
y = -3(1) - 4 = -3 - 4 = -7 \quad \text{(point (1, -7))}
\]
- When \( x = -1 \):
\[
y = -3(-1) - 4 = 3 - 4 = -1 \quad \text{(point (-1, -1))}
\]
Draw a dashed line for \( y = -3x - 4 \), since the inequality is greater than or equal to.
**Shade above this line**, because we want \( y \) to be greater than or equal to \( -3x - 4 \).
### Step 2: Graphing the second inequality
1. **Graph \( y = x + 3 \)**:
- The y-intercept is 3 (point (0, 3)).
- The slope is 1, which means for each increase of 1 in \( x \), \( y \) increases by 1.
Plot the y-intercept and use the slope to find another point:
- When \( x = 1 \):
\[
y = 1 + 3 = 4 \quad \text{(point (1, 4))}
\]
- When \( x = -1 \):
\[
y = -1 + 3 = 2 \quad \text{(point (-1, 2))}
\]
Draw a dashed line for \( y = x + 3 \), since the inequality is less than or equal to.
**Shade below this line**, as we want \( y \) to be less than or equal to \( x + 3 \).
### Step 3: Find the solution region
The solution set to the system of inequalities is the region where the shaded areas from both inequalities overlap.
1. The area above the line \( y = -3x - 4 \) and
2. The area below the line \( y = x + 3 \).
You can sketch this region on a coordinate graph paper or using graphing software to visualize the overlaps.
### Final Answer
To summarize, the solution region is the area where the shaded regions intersect, which represents all the points (x, y) that satisfy both inequalities.
1. The first inequality:
\[
-y \leq 3x + 4
\]
We can rewrite it as:
\[
y \geq -3x - 4
\]
2. The second inequality:
\[
-3x + 3y \leq 9
\]
We can simplify this by dividing the entire inequality by 3:
\[
-x + y \leq 3
\]
or
\[
y \leq x + 3
\]
Now we have the two inequalities to graph:
1. \( y \geq -3x - 4 \)
2. \( y \leq x + 3 \)
### Step 1: Graphing the first inequality
1. **Graph \( y = -3x - 4 \)**:
- The y-intercept is -4 (point (0, -4)).
- The slope is -3, which means for each increase of 1 in \( x \), \( y \) decreases by 3.
Plot the y-intercept and use the slope to find another point:
- When \( x = 1 \):
\[
y = -3(1) - 4 = -3 - 4 = -7 \quad \text{(point (1, -7))}
\]
- When \( x = -1 \):
\[
y = -3(-1) - 4 = 3 - 4 = -1 \quad \text{(point (-1, -1))}
\]
Draw a dashed line for \( y = -3x - 4 \), since the inequality is greater than or equal to.
**Shade above this line**, because we want \( y \) to be greater than or equal to \( -3x - 4 \).
### Step 2: Graphing the second inequality
1. **Graph \( y = x + 3 \)**:
- The y-intercept is 3 (point (0, 3)).
- The slope is 1, which means for each increase of 1 in \( x \), \( y \) increases by 1.
Plot the y-intercept and use the slope to find another point:
- When \( x = 1 \):
\[
y = 1 + 3 = 4 \quad \text{(point (1, 4))}
\]
- When \( x = -1 \):
\[
y = -1 + 3 = 2 \quad \text{(point (-1, 2))}
\]
Draw a dashed line for \( y = x + 3 \), since the inequality is less than or equal to.
**Shade below this line**, as we want \( y \) to be less than or equal to \( x + 3 \).
### Step 3: Find the solution region
The solution set to the system of inequalities is the region where the shaded areas from both inequalities overlap.
1. The area above the line \( y = -3x - 4 \) and
2. The area below the line \( y = x + 3 \).
You can sketch this region on a coordinate graph paper or using graphing software to visualize the overlaps.
### Final Answer
To summarize, the solution region is the area where the shaded regions intersect, which represents all the points (x, y) that satisfy both inequalities.