Question $|x+7|<6$

Studdy Solution

STEP 1

1. The inequality $|x + 7| < 6$ is an absolute value inequality.
2. We will need to consider the definition of absolute value to solve the inequality.

STEP 2

1. Understand the absolute value inequality.
2. Split the inequality into two separate inequalities.
3. Solve each inequality for $x$ .
4. Combine the solutions to find the range of $x$ .

STEP 3

The inequality $|x + 7| < 6$ means that the expression inside the absolute value, $x + 7$ , is less than 6 units away from 0 on the number line. This can be expressed as:
$-6 < x + 7 < 6$

STEP 4

Split the compound inequality into two separate inequalities:
1. $x + 7 > -6$
2. $x + 7 < 6$

STEP 5

Solve each inequality separately.
For the first inequality $x + 7 > -6$ :
Subtract 7 from both sides:
$x > -6 - 7$ $x > -13$
For the second inequality $x + 7 < 6$ :
Subtract 7 from both sides:
$x < 6 - 7$ $x < -1$

STEP 6

Combine the solutions from both inequalities to find the range of $x$ :
The solution is:
$-13 < x < -1$
This means $x$ is any number between $-13$ and $-1$ , not including $-13$ and $-1$ .
The solution to the inequality is:
$\boxed{(-13, -1)}$

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