Let's solve the inequality [tex]\( 5 - 3|6 - x| < 11 \)[/tex] step-by-step:
1. Isolate the absolute value term:
- Start with the given inequality: [tex]\( 5 - 3|6 - x| < 11 \)[/tex].
- Subtract 5 from both sides to isolate the absolute value term:
[tex]\[
-3|6 - x| < 6
\][/tex]
2. Divide both sides by -3:
- When dividing by a negative number, the direction of the inequality reverses. So:
[tex]\[
|6 - x| > -2
\][/tex]
3. Analyze the inequality [tex]\( |6 - x| > -2 \)[/tex]:
- The absolute value [tex]\( |6 - x| \)[/tex] represents a non-negative quantity (it is always greater than or equal to 0).
- Since the right side of the inequality is -2, which is negative, [tex]\( |6 - x| \)[/tex] will always be greater than -2 for any real number [tex]\( x \)[/tex].
4. Conclusion:
- Because an absolute value cannot be negative and thus is always greater than any negative number, the inequality [tex]\( |6 - x| > -2 \)[/tex] is always true for all real numbers [tex]\( x \)[/tex].
Therefore, the inequality [tex]\( 5 - 3|6 - x| < 11 \)[/tex] holds for all real numbers [tex]\( x \)[/tex].
