Question

1. graph y + 2 < 4x + 1

Explanation:

Step1: Rearrange the inequality

Subtract 2 from both sides of $y + 2<4x + 1$ to get $y<4x - 1$.

Step2: Identify the boundary - line

The boundary - line of the inequality $y<4x - 1$ is the equation $y = 4x-1$. Since the inequality is $y<4x - 1$ (not $y\leq4x - 1$), the boundary - line will be a dashed line.

Step3: Find the y - intercept

For the line $y = 4x-1$, when $x = 0$, $y=-1$. So the y - intercept is $(0,-1)$.

Step4: Find the slope

The slope of the line $y = 4x-1$ is $m = 4=\frac{4}{1}$. This means for every 1 unit increase in $x$, $y$ increases by 4 units.

Step5: Determine the shaded region

Choose a test point not on the line, say $(0,0)$. Substitute $x = 0$ and $y = 0$ into the inequality $y<4x - 1$. We get $0<4\times0 - 1$, or $0<-1$, which is false. So we shade the region that does not contain the point $(0,0)$.

To graph:

1. Plot the y - intercept $(0,-1)$.
2. Use the slope to find another point. From $(0,-1)$, move 1 unit to the right and 4 units up to get the point $(1,3)$.
3. Draw a dashed line through the points $(0,-1)$ and $(1,3)$.
4. Shade the region below the dashed line.

Answer:

Graph a dashed line $y = 4x-1$ (with y - intercept at $(0,-1)$ and slope 4) and shade the region below the line.