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Chapter 3: Problem 71

Explain why the equation \(\sqrt{x^{2}}=x\) is not valid for all real numbers \(x\) and should be replaced by the equation \(\sqrt{x^{2}}=|x|\).

Short Answer

Expert verified
The given equation \(\sqrt{x^{2}}=x\) is not valid for all real numbers \(x\) because it doesn't work for negative values of \(x\). Since squaring a number removes the sign and taking the square root of the square only guarantees a positive result or zero, we should use the equation \(\sqrt{x^{2}}=|x|\) to accommodate both positive and negative values of \(x\), making it valid for all real numbers.

Step by step solution

01

Examine the given equation for positive values of x

When \(x\) is a positive real number, the given equation works as expected. Let's take an example: Let, \(x = 3\). The equation \(\sqrt{x^{2}}=x\) becomes: \[ \sqrt{3^2} = 3 \\ 9 = 3^2 \\ \sqrt{9} = 3 \] Here, the given equation holds true.
02

Examine the given equation for negative values of x

Now, let's consider the case when \(x\) is a negative real number. Let \(x = -3\). The equation \(\sqrt{x^{2}}=x\) becomes: \[ \sqrt{(-3)^2} = -3 \\ 9 = (-3)^2 \\ \sqrt{9} = -3 \] Notice that this is not true, as \(\sqrt{9}\) should be positive. So, the given equation doesn't work for negative real numbers.
03

Explain why the correct equation is \(\sqrt{x^{2}}=|x|\)

To resolve the inconsistency we found earlier, we can use the equation \(\sqrt{x^{2}}=|x|\) which takes the absolute value of \(x\). This works because squaring a number removes the sign, and taking the square root of the square only guarantees a positive result or zero. Let's see if the equation works for a positive \(x\): Let \(x = 3\), the equation becomes: \[ \sqrt{3^2} = |3| \\ \sqrt{9} = 3 \\ \] And for a negative \(x\): Let \(x = -3\), the equation becomes: \[ \sqrt{(-3)^2} = |-3| \\ \sqrt{9} = 3 \\ \] As we can see, by replacing the given equation with \(\sqrt{x^{2}}=|x|\), the equation is valid for all real numbers, including both positive and negative values of \(x\).

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