Answer :
To solve the given system of inequalities:
[tex]\[ \left\{\begin{array}{l} y \leq 2x - 4 \\ y > -\frac{3}{4}x + 4 \end{array}\right. \][/tex]
we can follow these steps:
1. Graph the lines corresponding to the inequalities:
- For [tex]\( y = 2x - 4 \)[/tex]:
- This line has a slope of 2 and a y-intercept of -4.
- To plot the line, start at the point (0, -4) and use the slope to find another point. For example, if [tex]\( x = 1 \)[/tex], then [tex]\( y = 2(1) - 4 = -2 \)[/tex]. So, another point is (1, -2).
- For [tex]\( y = -\frac{3}{4}x + 4 \)[/tex]:
- This line has a slope of [tex]\(-\frac{3}{4}\)[/tex] and a y-intercept of 4.
- To plot the line, start at the point (0, 4) and use the slope to find another point. For example, if [tex]\( x = 4 \)[/tex], then [tex]\( y = -\frac{3}{4}(4) + 4 = 1 \)[/tex]. So, another point is (4, 1).
2. Determine the regions defined by the inequalities:
- For [tex]\( y \leq 2x - 4 \)[/tex]:
- This inequality includes the region on or below the line [tex]\( y = 2x - 4 \)[/tex].
- For [tex]\( y > -\frac{3}{4}x + 4 \)[/tex]:
- This inequality includes the region above the line [tex]\( y = -\frac{3}{4}x + 4 \)[/tex].
3. Find the intersection of the regions:
- To find the region that satisfies both inequalities, we need to find the overlapping area between the regions of the two inequalities.
4. Check the intersection of the lines:
- To find the intersection point of the lines [tex]\( y = 2x - 4 \)[/tex] and [tex]\( y = -\frac{3}{4}x + 4 \)[/tex], set the equations equal to each other:
[tex]\[ 2x - 4 = -\frac{3}{4}x + 4 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 2x + \frac{3}{4}x = 4 + 4 \][/tex]
[tex]\[ \frac{8x}{4} + \frac{3x}{4} = 8 \][/tex]
[tex]\[ \frac{11x}{4} = 8 \][/tex]
[tex]\[ 11x = 32 \][/tex]
[tex]\[ x = \frac{32}{11} \][/tex]
Now, find the corresponding [tex]\( y \)[/tex]-value by substituting [tex]\( x = \frac{32}{11} \)[/tex] into either equation. Using [tex]\( y = 2x - 4 \)[/tex]:
[tex]\[ y = 2\left(\frac{32}{11}\right) - 4 \][/tex]
[tex]\[ y = \frac{64}{11} - 4 \][/tex]
[tex]\[ y = \frac{64}{11} - \frac{44}{11} \][/tex]
[tex]\[ y = \frac{20}{11} \][/tex]
Therefore, the intersection point is [tex]\( \left( \frac{32}{11}, \frac{20}{11} \right) \)[/tex].
5. Draw the lines and the shaded regions:
- Graph the line [tex]\( y = 2x - 4 \)[/tex] and shade the region below and on the line.
- Graph the line [tex]\( y = -\frac{3}{4}x + 4 \)[/tex] and shade the region above the line.
6. Identify the solution region:
- The solution region is where the shaded regions overlap. This would generally be above the line [tex]\( y = -\frac{3}{4}x + 4 \)[/tex] and below the line [tex]\( y = 2x - 4 \)[/tex], limited to the intersection point [tex]\( \left( \frac{32}{11}, \frac{20}{11} \right) \)[/tex] and extending infinitely to the right.
Thus, the solution to the system of inequalities is the region that lies on or below the line [tex]\( y = 2x - 4 \)[/tex] and above the line [tex]\( y > -\frac{3}{4}x + 4 \)[/tex]. Mathematically:
[tex]\[ \left\{ (x, y) \mid y \leq 2x - 4 \ \text{and} \ y > -\frac{3}{4}x + 4 \right\} \][/tex]
This can be depicted clearly through graphing, with the intersection region indicating the solution set.
[tex]\[ \left\{\begin{array}{l} y \leq 2x - 4 \\ y > -\frac{3}{4}x + 4 \end{array}\right. \][/tex]
we can follow these steps:
1. Graph the lines corresponding to the inequalities:
- For [tex]\( y = 2x - 4 \)[/tex]:
- This line has a slope of 2 and a y-intercept of -4.
- To plot the line, start at the point (0, -4) and use the slope to find another point. For example, if [tex]\( x = 1 \)[/tex], then [tex]\( y = 2(1) - 4 = -2 \)[/tex]. So, another point is (1, -2).
- For [tex]\( y = -\frac{3}{4}x + 4 \)[/tex]:
- This line has a slope of [tex]\(-\frac{3}{4}\)[/tex] and a y-intercept of 4.
- To plot the line, start at the point (0, 4) and use the slope to find another point. For example, if [tex]\( x = 4 \)[/tex], then [tex]\( y = -\frac{3}{4}(4) + 4 = 1 \)[/tex]. So, another point is (4, 1).
2. Determine the regions defined by the inequalities:
- For [tex]\( y \leq 2x - 4 \)[/tex]:
- This inequality includes the region on or below the line [tex]\( y = 2x - 4 \)[/tex].
- For [tex]\( y > -\frac{3}{4}x + 4 \)[/tex]:
- This inequality includes the region above the line [tex]\( y = -\frac{3}{4}x + 4 \)[/tex].
3. Find the intersection of the regions:
- To find the region that satisfies both inequalities, we need to find the overlapping area between the regions of the two inequalities.
4. Check the intersection of the lines:
- To find the intersection point of the lines [tex]\( y = 2x - 4 \)[/tex] and [tex]\( y = -\frac{3}{4}x + 4 \)[/tex], set the equations equal to each other:
[tex]\[ 2x - 4 = -\frac{3}{4}x + 4 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 2x + \frac{3}{4}x = 4 + 4 \][/tex]
[tex]\[ \frac{8x}{4} + \frac{3x}{4} = 8 \][/tex]
[tex]\[ \frac{11x}{4} = 8 \][/tex]
[tex]\[ 11x = 32 \][/tex]
[tex]\[ x = \frac{32}{11} \][/tex]
Now, find the corresponding [tex]\( y \)[/tex]-value by substituting [tex]\( x = \frac{32}{11} \)[/tex] into either equation. Using [tex]\( y = 2x - 4 \)[/tex]:
[tex]\[ y = 2\left(\frac{32}{11}\right) - 4 \][/tex]
[tex]\[ y = \frac{64}{11} - 4 \][/tex]
[tex]\[ y = \frac{64}{11} - \frac{44}{11} \][/tex]
[tex]\[ y = \frac{20}{11} \][/tex]
Therefore, the intersection point is [tex]\( \left( \frac{32}{11}, \frac{20}{11} \right) \)[/tex].
5. Draw the lines and the shaded regions:
- Graph the line [tex]\( y = 2x - 4 \)[/tex] and shade the region below and on the line.
- Graph the line [tex]\( y = -\frac{3}{4}x + 4 \)[/tex] and shade the region above the line.
6. Identify the solution region:
- The solution region is where the shaded regions overlap. This would generally be above the line [tex]\( y = -\frac{3}{4}x + 4 \)[/tex] and below the line [tex]\( y = 2x - 4 \)[/tex], limited to the intersection point [tex]\( \left( \frac{32}{11}, \frac{20}{11} \right) \)[/tex] and extending infinitely to the right.
Thus, the solution to the system of inequalities is the region that lies on or below the line [tex]\( y = 2x - 4 \)[/tex] and above the line [tex]\( y > -\frac{3}{4}x + 4 \)[/tex]. Mathematically:
[tex]\[ \left\{ (x, y) \mid y \leq 2x - 4 \ \text{and} \ y > -\frac{3}{4}x + 4 \right\} \][/tex]
This can be depicted clearly through graphing, with the intersection region indicating the solution set.
