From the definition of absolute value, |3x| = 3x when x≥0 and |3x| = -3x when x<0.
The Correct Answer and Explanation is:
You’re correct in identifying how the absolute value function works. Let’s solve the expression step by step and then explain it clearly.
Correct Answer:
From the definition of absolute value:∣3x∣={3xif x≥0−3xif x<0|3x| = \begin{cases} 3x & \text{if } x \geq 0 \\ -3x & \text{if } x < 0 \end{cases}∣3x∣={3x−3xif x≥0if x<0
This is because the absolute value of a number is always non-negative. It represents the distance of that number from zero on the number line, regardless of direction.
Explanation
The absolute value of a number refers to its distance from zero on the number line. It is always a non-negative quantity. For any real number aaa, the absolute value is defined as:∣a∣={aif a≥0−aif a<0|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}∣a∣={a−aif a≥0if a<0
This means if the input is already positive or zero, the absolute value returns it unchanged. However, if the input is negative, the absolute value changes its sign to make it positive.
Now consider the expression ∣3x∣|3x|∣3x∣. The value of this expression depends on the sign of 3x3x3x, which in turn depends on the sign of xxx.
If x≥0x \geq 0x≥0, then 3x≥03x \geq 03x≥0. So the absolute value of 3x3x3x is simply 3x3x3x.
If x<0x < 0x<0, then 3x<03x < 03x<0. So the absolute value of 3x3x3x becomes −3x-3x−3x.
Therefore, we split the definition of ∣3x∣|3x|∣3x∣ based on whether xxx is greater than or equal to zero or less than zero:
- When x≥0x \geq 0x≥0, ∣3x∣=3x|3x| = 3x∣3x∣=3x
- When x<0x < 0x<0, ∣3x∣=−3x|3x| = -3x∣3x∣=−3x
This approach is consistent with the general definition of absolute value. It is important to analyze the sign of the expression inside the absolute value in order to evaluate it correctly. This technique is useful in solving equations and inequalities that involve absolute value terms.
