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'll first rewrite them into a more standard form suitable for graphing and then plot the lines corresponding to each inequality.
### Inequalities:
1. \(-y \leq 3x + 4\)
2. \(-3x + 3y \leq -9\)
### Step 1: Rewrite the Inequalities
1. For the first inequality, we can multiply through by \(-1\) (remembering to reverse the inequality symbol):
\[
y \geq -3x - 4
\]
2. For the second inequality:
\[
-3x + 3y \leq -9
\]
You can divide everything by \(3\) to simplify:
\[
-x + y \leq -3
\]
Again, multiplying by \(-1\) (and reversing the inequality):
\[
x - y \geq 3 \quad \text{or} \quad y \leq x - 3
\]
### Step 2: Graph the Inequalities
Now, we will graph the corresponding linear equations and then shade the appropriate regions.
#### Graph the Line \(y = -3x - 4\)
1. Identify the y-intercept: \(0, -4\) (point where the line crosses the y-axis).
2. The slope is \(-3\), so from \((0, -4)\), go down \(3\) and right \(1\) to get to \((1, -7)\).
3. Plot the points and draw the line. Since this is a \(\geq\) inequality, we will shade **above** the line.
#### Graph the Line \(y = x - 3\)
1. Identify the y-intercept: \(0, -3\).
2. The slope is \(1\), so from \((0, -3)\), go up \(1\) and right \(1\) to get to \((1, -2)\).
3. Plot the points and draw the line. Since this is a \(\leq\) inequality, we will shade **below** the line.
### Step 3: Determine the Solution Region
Now, find the overlapping shaded regions of the two inequalities:
- The first inequality \(y \geq -3x - 4\) will shade **above** the line \(y = -3x - 4\).
- The second inequality \(y \leq x - 3\) shades **below** the line \(y = x - 3\).
### Step 4: Identify the Solution
The solution to the system of inequalities is the region where the shaded areas overlap.
### Summary
- Shaded region above \(y = -3x - 4\)
- Shaded region below \(y = x - 3\)
**Note:** To visualize this accurately, drawing on a graph in a coordinate system is necessary to find the intersection points and any vertices of the region formed by the inequalities. This graphical solution will provide the exact area that satisfies both inequalities.
### Inequalities:
1. \(-y \leq 3x + 4\)
2. \(-3x + 3y \leq -9\)
### Step 1: Rewrite the Inequalities
1. For the first inequality, we can multiply through by \(-1\) (remembering to reverse the inequality symbol):
\[
y \geq -3x - 4
\]
2. For the second inequality:
\[
-3x + 3y \leq -9
\]
You can divide everything by \(3\) to simplify:
\[
-x + y \leq -3
\]
Again, multiplying by \(-1\) (and reversing the inequality):
\[
x - y \geq 3 \quad \text{or} \quad y \leq x - 3
\]
### Step 2: Graph the Inequalities
Now, we will graph the corresponding linear equations and then shade the appropriate regions.
#### Graph the Line \(y = -3x - 4\)
1. Identify the y-intercept: \(0, -4\) (point where the line crosses the y-axis).
2. The slope is \(-3\), so from \((0, -4)\), go down \(3\) and right \(1\) to get to \((1, -7)\).
3. Plot the points and draw the line. Since this is a \(\geq\) inequality, we will shade **above** the line.
#### Graph the Line \(y = x - 3\)
1. Identify the y-intercept: \(0, -3\).
2. The slope is \(1\), so from \((0, -3)\), go up \(1\) and right \(1\) to get to \((1, -2)\).
3. Plot the points and draw the line. Since this is a \(\leq\) inequality, we will shade **below** the line.
### Step 3: Determine the Solution Region
Now, find the overlapping shaded regions of the two inequalities:
- The first inequality \(y \geq -3x - 4\) will shade **above** the line \(y = -3x - 4\).
- The second inequality \(y \leq x - 3\) shades **below** the line \(y = x - 3\).
### Step 4: Identify the Solution
The solution to the system of inequalities is the region where the shaded areas overlap.
### Summary
- Shaded region above \(y = -3x - 4\)
- Shaded region below \(y = x - 3\)
**Note:** To visualize this accurately, drawing on a graph in a coordinate system is necessary to find the intersection points and any vertices of the region formed by the inequalities. This graphical solution will provide the exact area that satisfies both inequalities.