It sounds like you're working through an absolute value inequality problem. Let's break it down clearly and solve it step-by-step.
The inequality given is:
[tex]\[ |w + 4| > 6 \][/tex]
To solve this, we need to consider the definition of absolute values. For an inequality of the form [tex]\( |A| > C \)[/tex], where [tex]\( A \)[/tex] and [tex]\( C \)[/tex] are expressions, it implies:
[tex]\[ A < -C \quad \text{or} \quad A > C \][/tex]
Applying this to our specific problem:
[tex]\[ |w + 4| > 6 \][/tex]
we get:
[tex]\[ w + 4 < -6 \quad \text{or} \quad w + 4 > 6 \][/tex]
Now, let's solve each inequality separately:
### Solving [tex]\( w + 4 < -6 \)[/tex]
[tex]\[
w + 4 < -6
\][/tex]
Subtract 4 from both sides:
[tex]\[
w < -6 - 4
\][/tex]
[tex]\[
w < -10
\][/tex]
### Solving [tex]\( w + 4 > 6 \)[/tex]
[tex]\[
w + 4 > 6
\][/tex]
Subtract 4 from both sides:
[tex]\[
w > 6 - 4
\][/tex]
[tex]\[
w > 2
\][/tex]
### Combining the Solutions
The solutions to the original inequality [tex]\( |w + 4| > 6 \)[/tex] are:
[tex]\[
w < -10 \quad \text{or} \quad w > 2
\][/tex]
### Graphical Representation
On the number line, this can be represented as:
[tex]\[
\begin{array}{ccccccccc}
\cdots & -11 & \bullet & -10 & \cdots & 2 & \bullet & 3 & \cdots \\
\end{array}
\][/tex]
The filled bullets indicate the boundaries (-10 and 2) are not included in the solution set since the inequality is strict ([tex]\( > \)[/tex], not [tex]\( \geq \)[/tex]).
The solution essentially says that [tex]\( w \)[/tex] can be any value less than -10 or any value greater than 2.
Hence, the final answer is:
[tex]\[
w < -10 \quad \text{or} \quad w > 2
\][/tex]
