# How do you solve |t + 4| > 10?

Mar 3, 2018

tin(-∞,-14)U(6,∞)

#### Explanation:

Using the definition of the absolute value, we are able to split this inequality with absolute values into two without absolute values:

$t + 4 > 10$, when $t + 4 \ge 0$
$- \left(t + 4\right) > 10$, when $t + 4 < 0$

When solving each inequality separately, we get two conditions for t:

$t + 4 > 10$
$\left(t + 4\right) - 4 > 10 - 4$
$t > 6$

$- \left(t + 4\right) > 10$
$- t - 4 > 10$
$\left(- t - 4\right) + 4 > 10 + 4$
$- t > 14$
$- \left(- t\right) > - \left(14\right)$
$t < - 14$

We must then intersect each condition with its corresponding restriction:

$\left(t > 6\right)$$\left(t + 4 \ge 0\right)$
$\left(t > 6\right)$$\left(t \ge - 4\right)$
$t > 6$

$\left(t < - 14\right)$$\left(t + 4 < 10\right)$
$\left(t < - 14\right)$$\left(t < - 4\right)$
$t < - 14$

Lastly, we must union these two new conditions:

$\left(t > 6\right)$ U $\left(t < - 14\right)$
tin(-∞,-14)U(6,∞)

Mar 3, 2018

#### Explanation:

If you have $\left\mid x \right\mid > b$, then you can set it up as:

$x > b$ or $x < - b$

Therefore,

$\left\mid t + 4 \right\mid > 10$ becomes $t + 4 > 10$ or $t + 4 < - 10$

We solve each inequality.

$t + 4 > 10$

$t > 6$

$t + 4 < - 10$

$t < - 14$

$t$ either has to be greater than $6$ or less than $- 14$.