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Sagot :
Let's analyze each statement step-by-step in the context of the linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex]:
### Statement 1: "The slope of the line is -2."
- The given inequality is [tex]\( y > \frac{3}{4} x - 2 \)[/tex].
- The standard form for a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- Here, the slope [tex]\( m \)[/tex] is [tex]\(\frac{3}{4}\)[/tex], not -2.
- This statement is false.
### Statement 2: "The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line."
- For [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] represents the boundary.
- When graphing inequalities with a "greater than" (>) or "less than" (<), the line is dashed to indicate that points on the line are not included in the solution.
- This statement is true.
### Statement 3: "The area below the line is shaded."
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] indicates that we are interested in the region above the line because we want values of [tex]\( y \)[/tex] that are greater than what the line gives for corresponding [tex]\( x \)[/tex] values.
- Therefore, the area above the line should be shaded, not below.
- This statement is false.
### Statement 4: "One solution to the inequality is [tex]\( (0,0) \)[/tex]."
- To check if a point is a solution to the inequality, substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality.
- [tex]\( 0 > \frac{3}{4}(0) - 2 \implies 0 > -2 \)[/tex], which is true.
- Therefore, [tex]\( (0,0) \)[/tex] is indeed a solution.
- This statement is true.
### Statement 5: "The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex]."
- The [tex]\( y \)[/tex]-intercept of a line occurs where [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into [tex]\( y = \frac{3}{4} x - 2 \)[/tex], we get [tex]\( y = -2 \)[/tex].
- Therefore, the line intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0, -2) \)[/tex].
- This statement is true.
### Conclusion
True statements about the linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] 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].
So, the three true statements are:
1. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
2. One solution to the inequality is [tex]\( (0,0) \)[/tex].
3. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
### Statement 1: "The slope of the line is -2."
- The given inequality is [tex]\( y > \frac{3}{4} x - 2 \)[/tex].
- The standard form for a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- Here, the slope [tex]\( m \)[/tex] is [tex]\(\frac{3}{4}\)[/tex], not -2.
- This statement is false.
### Statement 2: "The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line."
- For [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] represents the boundary.
- When graphing inequalities with a "greater than" (>) or "less than" (<), the line is dashed to indicate that points on the line are not included in the solution.
- This statement is true.
### Statement 3: "The area below the line is shaded."
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] indicates that we are interested in the region above the line because we want values of [tex]\( y \)[/tex] that are greater than what the line gives for corresponding [tex]\( x \)[/tex] values.
- Therefore, the area above the line should be shaded, not below.
- This statement is false.
### Statement 4: "One solution to the inequality is [tex]\( (0,0) \)[/tex]."
- To check if a point is a solution to the inequality, substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality.
- [tex]\( 0 > \frac{3}{4}(0) - 2 \implies 0 > -2 \)[/tex], which is true.
- Therefore, [tex]\( (0,0) \)[/tex] is indeed a solution.
- This statement is true.
### Statement 5: "The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex]."
- The [tex]\( y \)[/tex]-intercept of a line occurs where [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into [tex]\( y = \frac{3}{4} x - 2 \)[/tex], we get [tex]\( y = -2 \)[/tex].
- Therefore, the line intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0, -2) \)[/tex].
- This statement is true.
### Conclusion
True statements about the linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] 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].
So, the three true statements are:
1. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
2. One solution to the inequality is [tex]\( (0,0) \)[/tex].
3. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
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