1)) Solve for \( w \). \( |-w|+1<9 \) or like \( \mathbf{x}<\mathbf{1} \) or \( \mathbf{x}>\mathbf{3} \). Use integers, proper fractions, or improper fractions in simplest form.

To solve the inequality \( |-w| + 1 < 9 \), we first isolate the absolute value. Subtracting 1 from both sides gives us \( |-w| < 8 \). The expression \( |-w| \) represents the distance of \( w \) from 0, so we can rewrite this as two inequalities: \[ -w < 8 \quad \text{and} \quad -w > -8. \] This translates to: \[ -w < 8 \implies w > -8 \] \[ -w > -8 \implies w < 8. \] Combining these two results gives us the solution: \[ -8 < w < 8. \] Now, let's summarize the final result: \( w \) is any number between \( -8 \) and \( 8 \), exclusive. Next, for the second part of your request, we can also draw a parallel with a similar concept. Consider you also have \( x < 1 \) or \( x > 3 \). It’s important to remember that these are disjoint intervals. Real numbers can be very flexible! So when you're finding solutions, sketch those number lines. It’ll help visualize how many solutions you really have, and it's quite fun to mark them! Lastly, when solving inequalities, it's easy to misread the signs. Be sure to flip the inequality signs when you multiply or divide both sides by a negative number. It’s a classic oops moment that can change your entire solution!