QuestionSolve the inequality $|4x + 6| + 3 > 25$ and express the solution set in interval notation.

Studdy Solution

STEP 1

Assumptions1. We are dealing with an absolute value inequality. . The inequality is $|4x+6|+3 >25$ .
3. We need to solve for $x$ .
4. The solution should be given in interval notation.

STEP 2

First, we need to isolate the absolute value expression. We can do this by subtracting from both sides of the inequality.
$|4x+6|+- >25-$

STEP 3

implify the inequality.
$|x+6| >22$

STEP 4

An absolute value inequality $|a| > b$ (where $b >0$ ) can be rewritten as a compound inequality $a < -b$ or $a > b$ . We can apply this to our inequality.
$4x+6 < -22 \quad or \quad4x+6 >22$

STEP 5

Now we have two inequalities to solve. Let's start with the first one, $4x+ < -22$ . We can solve for $x$ by first subtracting from both sides.
$4x+- < -22-$

STEP 6

implify the inequality.
$4x < -28$

STEP 7

Now, divide both sides by4 to solve for $x$ .
$\frac{4x}{4} < \frac{-28}{4}$

STEP 8

implify the inequality.
$x < -7$

STEP 9

Now let's solve the second inequality, $4x+6 >22$ . We can solve for $x$ by first subtracting6 from both sides.
$4x+6-6 >22-6$

STEP 10

implify the inequality.
$4x >16$

STEP 11

Now, divide both sides by4 to solve for $x$ .
$\frac{4x}{4} > \frac{16}{4}$

STEP 12

implify the inequality.
$x >4$

STEP 13

Now, we have two inequalities, $x < -7$ and $x >$ . In interval notation, this is represented as $(-\infty, -7) \cup (, \infty)$ .
The solution set is $(-\infty, -7) \cup (, \infty)$ .

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