Write the negation of the following statements: 'For every real number `x, x^(2) gt x`.'

Answer

To find the negation of the statement "For every real number \( x, x^2 > x \)", we can follow these steps:

### Step 1: Identify the Original Statement
The original statement is:
"For every real number \( x, x^2 > x \)."

### Step 2: Understand the Quantifier
The statement uses the universal quantifier "for every" (denoted as \( \forall \)). The negation of a statement that begins with "for every" is that there exists at least one case where the statement is not true.

### Step 3: Change the Quantifier
To negate the statement, we change "for every" to "there exists" (denoted as \( \exists \)). Thus, we rewrite the statement as:
"There exists a real number \( x \) such that..."

### Step 4: Negate the Inequality
Next, we need to negate the inequality \( x^2 > x \). The negation of \( x^2 > x \) is:
\( x^2 \leq x \).

### Step 5: Combine the Parts
Putting it all together, the negation of the original statement is:
"There exists a real number \( x \) such that \( x^2 \leq x \)."

### Final Answer
The negation of the statement "For every real number \( x, x^2 > x \)" is:
"There exists a real number \( x \) such that \( x^2 \leq x \)."

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