QuestionIdentify the mistake in these steps leading to the conclusion that if $x=1$ , then $2=1$ .

Studdy Solution

STEP 1

Assumptions1. We are given that $x=1$ . . We are to follow the logical progression of the conditionals to identify any mistakes.

STEP 2

First, let's evaluate the first conditional. If $x=1$ , then $x-2=-1$ . This is true, so there is no mistake in this conditional.
$x-2=-1$ $1-2=-1$ $-1=-1$

STEP 3

Next, let's evaluate the second conditional. If $x-2=-1$ , then $x^{2}+x-2=x^{2}-1$ . This is also true, so there is no mistake in this conditional.
$x^{2}+x-2=x^{2}-1$ $1^{2}+1-2=1^{2}-1$ $0=0$

STEP 4

Now, let's evaluate the third conditional. If $x^{2}+x-2=x^{2}-1$ , then $(x+2)(x-1)=(x+1)(x-1)$ . This is true, so there is no mistake in this conditional.
$(x+2)(x-1)=(x+1)(x-1)$ $(1+2)(1-1)=(1+1)(1-1)$ $0=0$

STEP 5

Next, let's evaluate the fourth conditional. If $(x+2)(x-1)=(x+1)(x-1)$ , then $(x+2)=(x+1)$ . This is where the mistake lies. We cannot divide by $x-1$ because $x-1=0$ , and division by zero is undefined. Therefore, this step is incorrect.

STEP 6

Since we have found the mistake, we do not need to evaluate the remaining conditionals. The mistake lies in the fourth conditional, where division by zero occurs.
The correct statement should be If $(x+2)(x-1)=(x+1)(x-1)$ , then $(x+2)=(x+1)$ , provided that $x-1 \neq0$ . However, since $x=1$ , $x-1=0$ , and the conditional is undefined.

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