Answer :
Let's analyze each statement about the linear inequality \( y > \frac{3}{4} x - 2 \) to determine which are true.
1. The slope of the line is -2.
- The given inequality is \( y > \frac{3}{4} x - 2 \).
- The standard form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Here, \( \frac{3}{4} \) is the coefficient of \( x \), so the slope is \( \frac{3}{4} \).
This statement is false.
2. The graph of \( y > \frac{3}{4} x - 2 \) is a dashed line.
- The inequality symbol \( > \) indicates that the line itself is not included in the solution set. Therefore, the boundary line should be dashed to signify that points on the line \( y = \frac{3}{4} x - 2 \) are not included in the solution.
This statement is true.
3. The area below the line is shaded.
- For the inequality \( y > \frac{3}{4} x - 2 \), we need the values of \( y \) that are greater than \( \frac{3}{4} x - 2 \).
- This means we shade the area above the line to represent values of \( y \) that satisfy the inequality.
This statement is false.
4. One solution to the inequality is \( (0,0) \).
- Let's test the point \( (0,0) \) in the inequality \( y > \frac{3}{4} x - 2 \).
- Substituting \( x = 0 \) and \( y = 0 \): \( 0 > \frac{3}{4}(0) - 2 \) simplifies to \( 0 > -2 \), which is true.
This statement is true.
5. The graph intercepts the \( y \)-axis at \( (0,-2) \).
- The y-intercept of the line \( y = \frac{3}{4} x - 2 \) occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation: \( y = \frac{3}{4}(0) - 2 = -2 \).
- Therefore, the y-intercept is \( (0, -2) \).
This statement is true.
To summarize, the three true statements about the given inequality \( y > \frac{3}{4} x - 2 \) are:
- The graph of \( y > \frac{3}{4} x - 2 \) is a dashed line.
- One solution to the inequality is \( (0,0) \).
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0, -2) \)[/tex].
1. The slope of the line is -2.
- The given inequality is \( y > \frac{3}{4} x - 2 \).
- The standard form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Here, \( \frac{3}{4} \) is the coefficient of \( x \), so the slope is \( \frac{3}{4} \).
This statement is false.
2. The graph of \( y > \frac{3}{4} x - 2 \) is a dashed line.
- The inequality symbol \( > \) indicates that the line itself is not included in the solution set. Therefore, the boundary line should be dashed to signify that points on the line \( y = \frac{3}{4} x - 2 \) are not included in the solution.
This statement is true.
3. The area below the line is shaded.
- For the inequality \( y > \frac{3}{4} x - 2 \), we need the values of \( y \) that are greater than \( \frac{3}{4} x - 2 \).
- This means we shade the area above the line to represent values of \( y \) that satisfy the inequality.
This statement is false.
4. One solution to the inequality is \( (0,0) \).
- Let's test the point \( (0,0) \) in the inequality \( y > \frac{3}{4} x - 2 \).
- Substituting \( x = 0 \) and \( y = 0 \): \( 0 > \frac{3}{4}(0) - 2 \) simplifies to \( 0 > -2 \), which is true.
This statement is true.
5. The graph intercepts the \( y \)-axis at \( (0,-2) \).
- The y-intercept of the line \( y = \frac{3}{4} x - 2 \) occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation: \( y = \frac{3}{4}(0) - 2 = -2 \).
- Therefore, the y-intercept is \( (0, -2) \).
This statement is true.
To summarize, the three true statements about the given inequality \( y > \frac{3}{4} x - 2 \) are:
- The graph of \( y > \frac{3}{4} x - 2 \) is a dashed line.
- One solution to the inequality is \( (0,0) \).
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0, -2) \)[/tex].
