To solve the inequality [tex]\( |x| - 9 > -8 \)[/tex], we can proceed with the following steps:
1. Simplify the Inequality:
[tex]\[
|x| - 9 > -8
\][/tex]
To simplify, we add 9 to both sides of the inequality:
[tex]\[
|x| - 9 + 9 > -8 + 9
\][/tex]
[tex]\[
|x| > 1
\][/tex]
2. Understand the Meaning of the Absolute Value Inequality:
The expression [tex]\( |x| > 1 \)[/tex] means that the distance between [tex]\( x \)[/tex] and 0 is greater than 1. This can be interpreted as two separate inequalities because the absolute value of a number is its distance from zero on the number line.
3. Break Down the Absolute Value Inequality:
Since [tex]\( |x| > 1 \)[/tex], we need to consider the two possible cases:
[tex]\[
x > 1
\][/tex]
and
[tex]\[
x < -1
\][/tex]
4. Combine the Results:
The solution to [tex]\( |x| > 1 \)[/tex] means that [tex]\( x \)[/tex] can be any value greater than 1 or any value less than -1. We can write this in interval notation for clarity:
[tex]\[
x \in (-\infty, -1) \cup (1, \infty)
\][/tex]
This represents the set of all values that satisfy the inequality [tex]\( |x| - 9 > -8 \)[/tex].
5. Conclusion:
The solution to the inequality [tex]\( |x| - 9 > -8 \)[/tex] is:
[tex]\[
x \in (-\infty, -1) \cup (1, \infty)
\][/tex]
In summary, [tex]\( x \)[/tex] must be either greater than 1 or less than -1 for the inequality [tex]\( |x| - 9 > -8 \)[/tex] to hold true.
