Question

1. where should the linear inequality $y \leq \frac{1}{4}x - 3$ be shaded?

graph of a line on a coordinate plane
explain how you came up with your answer:
a. above the line
b. below the line
c. neither

Explanation:

Step1: Analyze the inequality symbol

The inequality is \( y \leq \frac{1}{4}x - 3 \). The symbol "≤" means "less than or equal to". For a linear inequality in two variables, the inequality symbol tells us which side of the line to shade.

Step2: Recall the rule for shading

For an inequality of the form \( y \leq mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept), we shade below the line. This is because we are looking for all the points \((x,y)\) where the \( y \) - value is less than or equal to the value of the linear function \( \frac{1}{4}x - 3 \). If we take a test point, say the origin \((0,0)\), and substitute into the inequality: \( 0 \leq \frac{1}{4}(0)-3\) which simplifies to \( 0 \leq - 3 \), which is false. But if we take a point below the line, for example, \((0, - 4)\), substituting into the inequality gives \( - 4\leq\frac{1}{4}(0)-3\), or \( - 4\leq - 3 \), which is true. Also, the general rule for \( y \leq f(x) \) is to shade below the graph of \( y = f(x) \). Since our inequality is \( y \leq \frac{1}{4}x - 3 \), we shade below the line.

Answer:

B. Below the line