To solve the inequality [tex]\( |x| > -9 \)[/tex], we need to understand the properties of absolute values and inequalities.
1. Understanding Absolute Values:
- The absolute value of a number [tex]\( x \)[/tex], denoted [tex]\( |x| \)[/tex], is defined as the distance of [tex]\( x \)[/tex] from zero on the number line. Therefore, [tex]\( |x| \)[/tex] is always a non-negative number (i.e., [tex]\( |x| \geq 0 \)[/tex]).
- This means that the smallest value [tex]\( |x| \)[/tex] can take is 0. It can never be negative because distance cannot be negative.
2. Analyzing the Inequality [tex]\( |x| > -9 \)[/tex]:
- The inequality states that the absolute value of [tex]\( x \)[/tex] must be greater than -9.
- Since [tex]\( |x| \)[/tex] is always non-negative (i.e., greater than or equal to 0), it will always be greater than -9 because -9 is a negative number.
- In fact, any real number, when its absolute value is taken, will be greater than -9.
3. Conclusion:
- Because [tex]\( |x| \)[/tex] is always non-negative and non-negative values are always greater than any negative number, the inequality [tex]\( |x| > -9 \)[/tex] holds true for all real numbers [tex]\( x \)[/tex].
Therefore, the solution to the inequality [tex]\( |x| > -9 \)[/tex] is all real numbers. The inequality is always satisfied regardless of the value of [tex]\( x \)[/tex]. So, the correct solution is:
[tex]\[ \boxed{\text{all reals}} \][/tex]
