# How do you graph and solve 5 + |1-x/2| >=8?

Dec 21, 2015

$x \ge 8 \text{ or } x \le - 4$

#### Explanation:

1) Simplifying

First of all, bring $5$ to the other side. You can do so by subtracting $5$ on both sides of the inequality:

$5 + \left\mid 1 - \frac{x}{2} \right\mid \ge 8$

$\iff \left\mid 1 - \frac{x}{2} \right\mid \ge 3$

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2) Evaluating the absolute value function

To evaluate the absolute value function, we need to find out when $1 - \frac{x}{2} \ge 0$ holds and when $1 - \frac{x}{2} < 0$ holds.

To do this, let's find the point where $1 - \frac{x}{2} = 0$:

$1 - \frac{x}{2} = 0 \text{ " <=> " " x/2 = 1 " " <=> " } x = 2$

Plugging $x = 1$ and $x = 3$ gives:

$1 - \frac{x}{2} \ge 0 \textcolor{w h i t e}{\times x} \text{for } x \le 2$

$1 - \frac{x}{2} < 0 \textcolor{w h i t e}{\times x} \text{for } x > 2$

Now, you can evaluate the absolute value function:

 abs(1 - x/2) = { (color(white)(xx) 1 - x/2, color(white)(xxx) "for " 1 - x/2 >= 0 ), (-(1 - x/2), color(white)(xxx) "for " 1 - x/2 < 0) :}

 color(white)(xxxxx) = { (color(white)(x) 1 - x/2, color(white)(xxxxx) "for " x <= 2 ), (-1 + x/2, color(white)(xxxxx) "for " x > 2) :}

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3) Solving the two cases

3a) Let $x \le 2$.

This means that $1 - \frac{x}{2} \ge 0$ and $\left\mid 1 - \frac{x}{2} \right\mid \ge 1 - \frac{x}{2}$.

$\implies 1 - \frac{x}{2} \ge 3$

... subtract $1$ from both sides of the inequality...

$\iff - \frac{x}{2} \ge 2$

... multiply both sides with $- 2$.
Be careful: if multiplying with a negative number or dividing by a negative number, you need to flip the inequality sign!

$\iff x \le - 4$

Now, we need to combine the condition $x \le 2$ with the solution $x \le - 4$.
As $x \le - 4$ is the more restrictive condition, this is the solution for this case.

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3b) Let $x > 2$.

This means that $1 - \frac{x}{2} < 0$ and $\left\mid 1 - \frac{x}{2} \right\mid = - 1 + \frac{x}{2}$.

$\implies - 1 + \frac{x}{2} \ge 3$

... add $1$ to both sides of the inequality...

$\iff \frac{x}{2} \ge 4$

... multiply both sides of the inequality with $2$...

$\iff x \ge 8$

Between the two conditions, $x \ge 8$ and $x > 2$, the former is the more restrictive condition.

Thus, this is the solution for the second case.

In total, the solution is $x \ge 8 \text{ or } x \le - 4$

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4) Graphing

You can graph the absolute value function $\left\mid 1 - \frac{x}{2} \right\mid$ using the "elbow" of the absolute value function and the slope:

• the "elbow" is the point of the function where $1 - \frac{x}{2} = 0$ holds which is $x = 2$. Thus, the elbow is (2; 0).
• The slope is the factor of $\frac{x}{2}$, so it's $\frac{1}{2}$.

Thus the absolute function looks as follows:

graph{abs(1 - x/2) [-10, 10, -5, 5]}

The graph of

$\left\mid 1 - \frac{x}{2} \right\mid \ge 3$

is the part of the graph that is above the horizontal line at $y = 3$:

graph{(y - abs(1 - x/2))(y - 3) = 0 [-15, 15, -5, 10]}

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Hope that this helped! :-)