Answer :
Let's analyze the inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] and determine which statements about it are true.
1. The slope of the line is -2.
- The given inequality can be written as [tex]\( y > mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- For the inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{3}{4} \)[/tex].
- Therefore, the statement "The slope of the line is -2" is false.
2. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- Inequalities that do not include the equal sign (e.g., [tex]\( > \)[/tex] or [tex]\( < \)[/tex]) are represented with a dashed line to show that the points on the line are not included in the solution.
- Since [tex]\( y > \frac{3}{4} x - 2 \)[/tex] does not include the equal sign, the graph will indeed be a dashed line.
- Therefore, this statement is true.
3. The area below the line is shaded.
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] means that for any point [tex]\( (x, y) \)[/tex], [tex]\( y \)[/tex] must be greater than [tex]\( \frac{3}{4} x - 2 \)[/tex].
- This means the area above the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] is the solution region.
- Hence, the claim that the area below the line is shaded is false.
4. One solution to the inequality is [tex]\( (0,0) \)[/tex].
- To check if [tex]\( (0,0) \)[/tex] is a solution, substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[ 0 > \frac{3}{4} \cdot 0 - 2 \implies 0 > -2. \][/tex]
- The inequality [tex]\( 0 > -2 \)[/tex] is true.
- Thus, [tex]\( (0,0) \)[/tex] satisfies the inequality.
- Therefore, the statement "One solution to the inequality is [tex]\( (0,0) \)[/tex]" is true.
5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
- The equation of the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] can be used to determine the y-intercept where [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{3}{4} \cdot 0 - 2 = -2. \][/tex]
- Hence, the y-intercept is [tex]\( (0, -2) \)[/tex].
- Therefore, the statement "The graph intercepts the y-axis at [tex]\( (0,-2) \)[/tex]" is true.
Summarizing the truths:
- The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line. (True)
- One solution to the inequality is [tex]\( (0,0) \)[/tex]. (True)
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex]. (True)
Thus, the three true statements are:
- The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- One solution to the inequality is [tex]\( (0,0) \)[/tex].
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
1. The slope of the line is -2.
- The given inequality can be written as [tex]\( y > mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- For the inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{3}{4} \)[/tex].
- Therefore, the statement "The slope of the line is -2" is false.
2. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- Inequalities that do not include the equal sign (e.g., [tex]\( > \)[/tex] or [tex]\( < \)[/tex]) are represented with a dashed line to show that the points on the line are not included in the solution.
- Since [tex]\( y > \frac{3}{4} x - 2 \)[/tex] does not include the equal sign, the graph will indeed be a dashed line.
- Therefore, this statement is true.
3. The area below the line is shaded.
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] means that for any point [tex]\( (x, y) \)[/tex], [tex]\( y \)[/tex] must be greater than [tex]\( \frac{3}{4} x - 2 \)[/tex].
- This means the area above the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] is the solution region.
- Hence, the claim that the area below the line is shaded is false.
4. One solution to the inequality is [tex]\( (0,0) \)[/tex].
- To check if [tex]\( (0,0) \)[/tex] is a solution, substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[ 0 > \frac{3}{4} \cdot 0 - 2 \implies 0 > -2. \][/tex]
- The inequality [tex]\( 0 > -2 \)[/tex] is true.
- Thus, [tex]\( (0,0) \)[/tex] satisfies the inequality.
- Therefore, the statement "One solution to the inequality is [tex]\( (0,0) \)[/tex]" is true.
5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
- The equation of the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] can be used to determine the y-intercept where [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \frac{3}{4} \cdot 0 - 2 = -2. \][/tex]
- Hence, the y-intercept is [tex]\( (0, -2) \)[/tex].
- Therefore, the statement "The graph intercepts the y-axis at [tex]\( (0,-2) \)[/tex]" is true.
Summarizing the truths:
- The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line. (True)
- One solution to the inequality is [tex]\( (0,0) \)[/tex]. (True)
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex]. (True)
Thus, the three true statements are:
- The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- One solution to the inequality is [tex]\( (0,0) \)[/tex].
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
