Answer :
To determine which statements are true about the linear inequality \( y > \frac{3}{4} x - 2 \), let's analyze it step-by-step:
1. The slope of the line is -2.
- This statement is false. The slope of the line is given by the coefficient of \( x \) in the inequality \( y = \frac{3}{4} x - 2 \). Therefore, the correct slope is \(\frac{3}{4}\), not -2.
2. The graph of \( y > \frac{3}{4} x - 2 \) is a dashed line.
- This statement is true. The inequality is a strict inequality (greater than) which means the line itself is not included in the solution set. Thus, it is represented by a dashed line.
3. The area below the line is shaded.
- This statement is false. For the inequality \( y > \frac{3}{4} x - 2 \), the region above the line is shaded because the inequality is "greater than" \( \frac{3}{4} x - 2 \).
4. One solution to the inequality is \((0,0)\).
- This statement is true. To check if \((0,0)\) is a solution, substitute \( x = 0 \) and \( y = 0 \) into the inequality:
[tex]\[ 0 > \frac{3}{4}(0) - 2 \implies 0 > -2 \][/tex]
Since this statement is true, \((0,0)\) is indeed a solution.
5. The graph intercepts the y-axis at \((0,-2)\).
- This statement is true. The y-axis intercept occurs where \( x = 0 \). Substituting \( x = 0 \) into the equation \( y = \frac{3}{4} x - 2 \) yields:
[tex]\[ y = \frac{3}{4}(0) - 2 = -2 \][/tex]
Therefore, the y-intercept is at \((0, -2)\).
Based on this analysis, the three correct options 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 y-axis at [tex]\((0,-2)\)[/tex].
1. The slope of the line is -2.
- This statement is false. The slope of the line is given by the coefficient of \( x \) in the inequality \( y = \frac{3}{4} x - 2 \). Therefore, the correct slope is \(\frac{3}{4}\), not -2.
2. The graph of \( y > \frac{3}{4} x - 2 \) is a dashed line.
- This statement is true. The inequality is a strict inequality (greater than) which means the line itself is not included in the solution set. Thus, it is represented by a dashed line.
3. The area below the line is shaded.
- This statement is false. For the inequality \( y > \frac{3}{4} x - 2 \), the region above the line is shaded because the inequality is "greater than" \( \frac{3}{4} x - 2 \).
4. One solution to the inequality is \((0,0)\).
- This statement is true. To check if \((0,0)\) is a solution, substitute \( x = 0 \) and \( y = 0 \) into the inequality:
[tex]\[ 0 > \frac{3}{4}(0) - 2 \implies 0 > -2 \][/tex]
Since this statement is true, \((0,0)\) is indeed a solution.
5. The graph intercepts the y-axis at \((0,-2)\).
- This statement is true. The y-axis intercept occurs where \( x = 0 \). Substituting \( x = 0 \) into the equation \( y = \frac{3}{4} x - 2 \) yields:
[tex]\[ y = \frac{3}{4}(0) - 2 = -2 \][/tex]
Therefore, the y-intercept is at \((0, -2)\).
Based on this analysis, the three correct options 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 y-axis at [tex]\((0,-2)\)[/tex].
