Answer :
To graph the solution to the system of inequalities [tex]\( y > 4x + 3 \)[/tex] and [tex]\( y \geq -3x + 8 \)[/tex], follow these steps:
### Step 1: Graph Each Inequality Separately
1. Graph the line [tex]\( y = 4x + 3 \)[/tex]:
- This line has a slope of 4 and a y-intercept of 3.
- Plot the y-intercept (0, 3).
- From the y-intercept, use the slope to find another point: rise 4 units and run 1 unit to the right, which gives the point (1, 7).
- Draw a dashed line through these points because the inequality is strict ([tex]\( y > 4x + 3 \)[/tex]).
2. Graph the line [tex]\( y = -3x + 8 \)[/tex]:
- This line has a slope of -3 and a y-intercept of 8.
- Plot the y-intercept (0, 8).
- From the y-intercept, use the slope to find another point: fall 3 units and run 1 unit to the right, which gives the point (1, 5).
- Draw a solid line through these points because the inequality is non-strict ([tex]\( y \geq -3x + 8 \)[/tex]).
### Step 2: Determine the Shaded Regions
1. For [tex]\( y > 4x + 3 \)[/tex]:
- Shade the area above the dashed line [tex]\( y = 4x + 3 \)[/tex].
2. For [tex]\( y \geq -3x + 8 \)[/tex]:
- Shade the area above the solid line [tex]\( y = -3x + 8 \)[/tex].
### Step 3: Find the Intersection of the Shaded Regions
The solution to the system of inequalities is where the shaded regions of both inequalities overlap.
- For [tex]\( y > 4x + 3 \)[/tex], you shade above the dashed line [tex]\( y = 4x + 3 \)[/tex].
- For [tex]\( y \geq -3x + 8 \)[/tex], you shade above the solid line [tex]\( y = -3x + 8 \)[/tex].
### Step 4: Plot the Graph
1. Draw the x-axis and y-axis.
2. Plot the lines for the inequalities:
- Draw the dashed line for [tex]\( y = 4x + 3 \)[/tex].
- Draw the solid line for [tex]\( y = -3x + 8 \)[/tex].
3. Shade the appropriate regions:
- Shade above the dashed line [tex]\( y = 4x + 3 \)[/tex].
- Shade above the solid line [tex]\( y = -3x + 8 \)[/tex].
4. Identify the intersection:
- The solution region is where the shaded areas overlap. This region is bounded above by [tex]\( y \geq -3x + 8 \)[/tex] and below by [tex]\( y > 4x + 3 \)[/tex].
### Sample Graph Description
- If plotted correctly, the dashed line [tex]\( y = 4x + 3 \)[/tex] will intersect the solid line [tex]\( y = -3x + 8 \)[/tex] at their intersection point.
- Locate the point of intersection by solving the equations [tex]\( 4x + 3 = -3x + 8 \)[/tex] for x:
[tex]\[ 4x + 3 = -3x + 8 \\ 7x = 5 \\ x = \frac{5}{7} \][/tex]
- Substitute [tex]\( x = \frac{5}{7} \)[/tex] into one of the equations to find y:
[tex]\[ y = 4\left(\frac{5}{7}\right) + 3 = \frac{20}{7} + 3 = \frac{20}{7} + \frac{21}{7} = \frac{41}{7} \][/tex]
- Thus, the point of intersection is [tex]\( \left(\frac{5}{7}, \frac{41}{7}\right) \)[/tex].
The final graph includes the dashed and solid lines, and the solution region is the area between these lines above [tex]\( y = 4x + 3 \)[/tex] and below or on [tex]\( y \geq -3x + 8 \)[/tex].
### Step 1: Graph Each Inequality Separately
1. Graph the line [tex]\( y = 4x + 3 \)[/tex]:
- This line has a slope of 4 and a y-intercept of 3.
- Plot the y-intercept (0, 3).
- From the y-intercept, use the slope to find another point: rise 4 units and run 1 unit to the right, which gives the point (1, 7).
- Draw a dashed line through these points because the inequality is strict ([tex]\( y > 4x + 3 \)[/tex]).
2. Graph the line [tex]\( y = -3x + 8 \)[/tex]:
- This line has a slope of -3 and a y-intercept of 8.
- Plot the y-intercept (0, 8).
- From the y-intercept, use the slope to find another point: fall 3 units and run 1 unit to the right, which gives the point (1, 5).
- Draw a solid line through these points because the inequality is non-strict ([tex]\( y \geq -3x + 8 \)[/tex]).
### Step 2: Determine the Shaded Regions
1. For [tex]\( y > 4x + 3 \)[/tex]:
- Shade the area above the dashed line [tex]\( y = 4x + 3 \)[/tex].
2. For [tex]\( y \geq -3x + 8 \)[/tex]:
- Shade the area above the solid line [tex]\( y = -3x + 8 \)[/tex].
### Step 3: Find the Intersection of the Shaded Regions
The solution to the system of inequalities is where the shaded regions of both inequalities overlap.
- For [tex]\( y > 4x + 3 \)[/tex], you shade above the dashed line [tex]\( y = 4x + 3 \)[/tex].
- For [tex]\( y \geq -3x + 8 \)[/tex], you shade above the solid line [tex]\( y = -3x + 8 \)[/tex].
### Step 4: Plot the Graph
1. Draw the x-axis and y-axis.
2. Plot the lines for the inequalities:
- Draw the dashed line for [tex]\( y = 4x + 3 \)[/tex].
- Draw the solid line for [tex]\( y = -3x + 8 \)[/tex].
3. Shade the appropriate regions:
- Shade above the dashed line [tex]\( y = 4x + 3 \)[/tex].
- Shade above the solid line [tex]\( y = -3x + 8 \)[/tex].
4. Identify the intersection:
- The solution region is where the shaded areas overlap. This region is bounded above by [tex]\( y \geq -3x + 8 \)[/tex] and below by [tex]\( y > 4x + 3 \)[/tex].
### Sample Graph Description
- If plotted correctly, the dashed line [tex]\( y = 4x + 3 \)[/tex] will intersect the solid line [tex]\( y = -3x + 8 \)[/tex] at their intersection point.
- Locate the point of intersection by solving the equations [tex]\( 4x + 3 = -3x + 8 \)[/tex] for x:
[tex]\[ 4x + 3 = -3x + 8 \\ 7x = 5 \\ x = \frac{5}{7} \][/tex]
- Substitute [tex]\( x = \frac{5}{7} \)[/tex] into one of the equations to find y:
[tex]\[ y = 4\left(\frac{5}{7}\right) + 3 = \frac{20}{7} + 3 = \frac{20}{7} + \frac{21}{7} = \frac{41}{7} \][/tex]
- Thus, the point of intersection is [tex]\( \left(\frac{5}{7}, \frac{41}{7}\right) \)[/tex].
The final graph includes the dashed and solid lines, and the solution region is the area between these lines above [tex]\( y = 4x + 3 \)[/tex] and below or on [tex]\( y \geq -3x + 8 \)[/tex].
