Question

inequalities
which statements are true about the linear inequality ( y > \frac{3}{4}x - 2 )? choose three correct answers.

• the graph of ( y > \frac{3}{4}x - 2 ) is a dashed line.
• the slope of the line is ( -2 ).
• the graph intersects the ( y )-axis at ( (0, -2) ).
• one solution to the inequality is ( (0, 0) ).
• the area below the line is shaded.

Explanation:

1. For the statement "The graph of \( y > \frac{3}{4}x - 2 \) is a dashed line": In linear inequalities, if the inequality is strict (\(>\) or \(<\)), the boundary line is dashed (since the points on the line are not included). So this is true.
2. For the statement "The slope of the line is \(-2\)": The equation of the line is in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For \(y=\frac{3}{4}x - 2\), the slope \(m=\frac{3}{4}\), not \(-2\). So this is false.
3. For the statement "The graph intersects the \(y\) - axis at \((0,-2)\)": To find the \(y\) - intercept, we set \(x = 0\) in the equation \(y=\frac{3}{4}x-2\). When \(x = 0\), \(y=\frac{3}{4}(0)-2=-2\). So the line intersects the \(y\) - axis at \((0, - 2)\), and this is true.
4. For the statement "One solution to the inequality is \((0,0)\)": Substitute \(x = 0\) and \(y = 0\) into the inequality \(y>\frac{3}{4}x - 2\). We get \(0>\frac{3}{4}(0)-2\), which simplifies to \(0>-2\), which is true. So \((0,0)\) is a solution.
5. For the statement "The area below the line is shaded": The inequality is \(y>\frac{3}{4}x - 2\), which means we shade the area above the line (since \(y\) values are greater than the line's \(y\) values for a given \(x\)). So this is false.

Answer:

• The graph of \( y > \frac{3}{4}x - 2 \) is a dashed line.
• The graph intersects the \(y\) - axis at \((0,-2)\).
• One solution to the inequality is \((0,0)\).