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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.

# Algebra Missteps

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?

1. $$\,\,\,\,\,\sqrt{x - 5} = -3$$

2. $$\,\,\,\,\,x - 5 = 9$$

3. $$\,\,\,\,\,x = 14$$

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

1. $$\,\,\,\,\,\sqrt{x} + \sqrt{1} = 1$$

2. $$\,\,\,\,\,\sqrt{x} + 1 = 1$$

3. $$\,\,\,\,\,\sqrt{x} = 0$$

4. $$\,\,\,\,\,x = 0$$

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