Replace the symbol □with either = or ≠to make the resulting statement true for all real numbers

step 1 All right, this problem is asking you to replace the square symbol there with either equal to or

Replace the symbol □with either = or ≠to make the resulting...

Replace the symbol $\square$ with either $=$ or $\neq$ to make the resulting statement true for all real numbers $a, b$ $c,$ and $d,$ whenever the expressions are defined.
$$\frac{a b+a c}{a} \square b+a c$$

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Key Concepts

Algebraic Simplification

This concept involves rewriting expressions in a simpler or more manageable form by combining like terms, factoring, or eliminating common factors. In the context of the problem, it is used to simplify the fraction by factoring and canceling common factors in the numerator and denominator.

Cancellation Law in Fractions

This principle states that a multiplicative factor common to both the numerator and the denominator of a fraction can be cancelled out, provided the factor is not zero. In the given expression, cancelling the common factor (assuming it is nonzero) transforms the fraction into a simpler expression, which directly leads to the equality.

Domain Restrictions in Rational Expressions

When dealing with rational expressions, it is important to consider the values for which the expression is defined. Specifically, any variable in the denominator must not be zero to avoid undefined expressions. This concept ensures that the algebraic manipulations, such as cancellation, are valid only when the denominator is nonzero.

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