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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: \[\begin{align} \sqrt{x+ 4} &= x -2\\ x+ 4 &= (x -2)^2\\ x+ 4 &= x^2 - 4x + 4\\ 0 &= x^2 - 5x\\ 0 &= x(x - 5). \end{align}\]

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

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

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Is the following solution valid?

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

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

  3. \(\,\,\,\,\,x = 14\)

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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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