Does the following equation have a true or false value?
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Does the following equation have a true or false value?
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: The numerator can be factored as a difference of squares: .
Step 2: Substitute this factorization into the equation:
.
Step 3: Simplify the fraction by canceling the common terms, giving , which is always true, except where the expression is undefined.
Step 4: The expression is undefined when the denominator is zero, i.e., when or . Thus, and .
In conclusion, the given equation is True only when .
True only when .
Solve:
\( (2+x)(2-x)=0 \)
Great question! While the simplified form 1 = 1 is always true, you must remember the original expression had restrictions. The fraction doesn't exist when x = ±9!
At these values, both the numerator and denominator equal zero, creating the indeterminate form . This means the expression is undefined at these points, even though it simplifies to 1 everywhere else.
Look at the original denominator before any simplification. Set each factor equal to zero and solve:
These are your restricted values!
Exactly! This creates holes in the graph at x = ±9. The function equals 1 everywhere except these two points, where it's undefined. Think of it as a horizontal line y = 1 with two holes punched out.
This is a trick question! It's neither simply true nor false. The correct answer is "True only when x ≠ ±9" because you must account for the domain restrictions of the original expression.
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