# How do you graph y+4<=1/2(x-4)?

Sep 2, 2017

See below.

#### Explanation:

Subtract 4 from both sides to get $y \le \frac{1}{2} \left(x - 4\right) - 4$
Simplify right side to get $y \le \frac{1}{2} x - 6$

Because this is also an equal to inequality we can say that

$y = \frac{1}{2} x - 6$

This gives us the equation of a line that can be plotted.
All values that lie on the line are included values. The shaded area of included values can be found by taking a set coordinates of from either side of the line and checking them in the inequality to see if they satisfy it.

See graph:
graph{y + 4 <= (1/2)(x-4) [-25.67, 25.66, -12.83, 12.84]}

Sep 2, 2017

#### Explanation:

As $y + 4 \le \frac{1}{2} \left(x - 4\right)$. we have

$y \le \frac{1}{2} x - 2 - 4$ or $y \le \frac{1}{2} x - 6$

Hence first draw te graph of $y = \frac{1}{2} x - 6$

When $x = 0$, we have $y = - 6$ and when $y = 0$, $x = 12$

Hence we can draw the graph of $y = \frac{1}{2} x - 6$ by joining poiints $\left(0 , - 6\right)$ and $\left(12 , 0\right)$. The graph appears as follows:

graph{(x-2y-12)(x^2+(y+6)^2-0.02)((x-12)^2+y^2-0.02)=0 [-4.46, 15.54, -7.28, 2.72]}

This divides the Cartesian plane in three parts,

One - on the line - all these points satisfy the inequality $y + 4 \le \frac{1}{2} \left(x - 4\right)$ as on the line $y + 4 = \frac{1}{2} \left(x - 4\right)$. Observe that $y + 4 \le \frac{1}{2} \left(x - 4\right)$ has equality sign.

Two - to the left of line. Let us pick up the point $\left(0 , 0\right)$ as for this $0 + 4 \le \frac{1}{2} \left(0 - 4\right)$ or $4 \le - 2$ and this does not satisfy $y + 4 \le \frac{1}{2} \left(x - 4\right)$

Three - to the right of line. Let us pick up $\left(10 , - 5\right)$ and at this we have $- 5 + 4 \le \frac{1}{2} \left(10 - 4\right)$ or $- 1 \le 3$ and this does satisfy $y + 4 \le \frac{1}{2} \left(x - 4\right)$.

As the line and portion of the plane to the right of it satisfies the inequality $y + 4 \le \frac{1}{2} \left(x - 4\right)$, the graph apears as follows:

graph{y+4 <= 1/2(x-4) [-4.46, 15.54, -7.28, 2.72]}

Note that if we had $y + 4 < \frac{1}{2} \left(x - 4\right)$, only the third portion would have satisfied the equation. As points on line then do not satisfy, we draw it as dotted as shown below.

graph{y+4 < 1/2(x-4) [-4.46, 15.54, -7.28, 2.72]}