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Chapter 9: Problem 46

Explain why \(|4 l+9| \leq-10\) has no solution.

Short Answer

Expert verified
The given inequality \( |4l + 9| \leq -10 \) has no solution because the absolute value function is always non-negative, and the right side of the inequality is a constant negative number (-10). Thus, the left side of the inequality will never be negative, and no real number (l) can satisfy the inequality.

Step by step solution

01

Understanding of absolute value

The absolute value of a real number is its distance from zero on the number line. It is defined as follows: \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \). From this definition, we can see that the absolute value function is always non-negative, meaning that for any real number x, \( |x| \geq 0 \).
02

Analyzing the given inequality

Now, let's analyze the given inequality \( |4l + 9| \leq -10 \). We can make the following observations: 1. The left side of the inequality is an absolute value function, which is always non-negative. 2. The right side of the inequality is a constant negative number, -10.
03

Reaching a conclusion

Since the absolute value function is always non-negative, no real number (l) can satisfy the inequality \( |4l + 9| \leq -10 \), as the left side will never be negative. Therefore, the given inequality has no solution.

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Most popular questions from this chapter

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