Certainly! Let's solve the inequality [tex]\( |x| + 7 < 4 \)[/tex] step by step.
1. Isolate the absolute value term:
[tex]\[
|x| + 7 < 4
\][/tex]
2. Subtract 7 from both sides:
[tex]\[
|x| < 4 - 7
\][/tex]
3. Simplify the right side:
[tex]\[
|x| < -3
\][/tex]
4. Interpret the result:
The expression [tex]\( |x| < -3 \)[/tex] seems problematic because the absolute value of any real number is always non-negative (i.e., [tex]\( |x| \geq 0 \)[/tex] for all [tex]\( x \)[/tex]). There is no real number [tex]\( x \)[/tex] such that its absolute value is less than a negative number. In this case, [tex]\( -3 \)[/tex] is a negative number, and thus the inequality [tex]\( |x| < -3 \)[/tex] is impossible to satisfy for any real number.
Therefore, the solution set for the inequality [tex]\( |x| + 7 < 4 \)[/tex] is the empty set, denoted by [tex]\( \varnothing \)[/tex].
