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# Common Misconceptions (Algebra)

Arm yourself with the tools to be the king or queen of heady mathematical debates, like the age old question of whether 0.999.... = 1.

What are the solution(s) to the equation \[\sqrt{x+ 4} = x -2?\]

(Partial solution)

The equation can be rewritten as follows:
\[\begin{align}
\sqrt{x+ 4} &= x -2\\
x+ 4 &= (x -2)^2\\
x+ 4 &= x^2 - 4x + 4\\
0 &= x^2 - 5x\\
0 &= x(x - 5).
\end{align}\]

Where is the error in the solution to this problem?

Find the solution set of \(20 - x > 5\).

Line 1. \(\,\,\,\,\,20 - x > 5\)

Line 2. \(\,\,\,\,\,-x > -15\)

Line 3. \(\,\,\,\,\,x >15\)

Where is the error in the solution to this problem?

Find the solution set of \(3 - x < 8\).

Step 1. \(\,\,\,\,\,3 - x < 8\)

Step 2. \(\,\,\,\,\,-x < 5\)

Step 3. \(\,\,\,\,\,x < -5\)

Is the following solution valid?

\(\,\,\,\,\,\sqrt{x - 5} = -3\)

\(\,\,\,\,\,x - 5 = 9\)

\(\,\,\,\,\,x = 14\)

Consider the following work done to solve the equation \[\sqrt{x + 1} = 1.\]

\(\,\,\,\,\,\sqrt{x} + \sqrt{1} = 1\)

\(\,\,\,\,\,\sqrt{x} + 1 = 1\)

\(\,\,\,\,\,\sqrt{x} = 0\)

\(\,\,\,\,\,x = 0\)

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