Solve $\left|\frac{x+3}{5}\right| \leq-4$
A) All real numbers
B) $-23 \leq x \leq 17$
C) $x \leq-23$ and $x \geq 17$
D) No solution
Answer (1)
Recognize that the absolute value of any expression is always non-negative.
Note that 5 x + 3 ≥ 0 for all real numbers x .
Since the absolute value is always non-negative, it can never be less than or equal to a negative number like -4.
Conclude that there is no solution to the inequality: No solution .
Explanation
Understanding the Problem We are given the inequality 5 x + 3 ≤ − 4 . We need to find the solution set for x . The absolute value of any real number is non-negative.
Absolute Value Property The absolute value of any real number is always non-negative. This means that 5 x + 3 ≥ 0 for all real numbers x .
No Solution Since the absolute value is always non-negative, it can never be less than or equal to a negative number like -4. Therefore, the inequality 5 x + 3 ≤ − 4 has no solution.
Final Answer The solution to the inequality 5 x + 3 ≤ − 4 is no solution.
Examples
Absolute value inequalities can be used in real life to determine acceptable ranges of error in measurements or manufacturing. For example, if a machine is supposed to cut a metal rod to 10 cm, an absolute value inequality can define the acceptable deviation from this length. If the acceptable deviation is 0.1 cm, the actual length x must satisfy ∣ x − 10∣ l e q 0.1 . However, in this case, the absolute value is less than or equal to a negative number, which is impossible, indicating that there is no acceptable deviation.
