For all values of and for which the expression is defined, the rational expression is equivalent to which of the following?

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SAT Equivalent Expressions Question 2 | Free SAT Prep | Aniko

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Question 2·Medium·Equivalent Expressions

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For all values of $x$ and $y$ for which the expression is defined, the rational expression

\frac{4x^2 - 9y^2}{2x + 3y}

is equivalent to which of the following?

For rational expression equivalence questions, first look for factoring opportunities in the numerator and denominator, especially special products like differences of squares ( $a^2 - b^2$ ), perfect-square trinomials, or common monomial factors. Write both numerator and denominator in fully factored form, then cancel only entire common factors (like $2x + 3y$ ), never individual terms within sums or differences. Keep in mind that any values making the original denominator zero are excluded from the domain, so simplification is valid only for the remaining values. Avoid trying to "distribute" division over addition or subtraction; if the denominator is a sum, factoring is usually the safe and fast route.

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Focus on the numerator

Look at $4x^2 - 9y^2$ . Can you rewrite each term as a square of something simpler (like $(2x)^2$ or $(3y)^2$ )?

Recall a factoring pattern

There is a standard formula $a^2 - b^2 = (a - b)(a + b)$ . What choices of $a$ and $b$ will make this formula match $4x^2 - 9y^2$ ?

Use cancellation correctly

After you factor the numerator, write the entire fraction with the factored form. Do you see a factor that appears both in the numerator and denominator? What happens when you divide the same nonzero factor from the top and bottom of a fraction?

Desmos Guide

Enter the original expression with a parameter

In Desmos, type (4x^2 - 9y^2)/(2x + 3y) on one line. When Desmos prompts you, create a slider for y so that y is treated as a constant and x is the variable on the graph.

Graph each answer choice

On separate lines, type each option using the same y slider, for example: 2x + 3y, 2x - (3/2)y, 2x + (3/2)y, and 2x - 3y. Desmos will draw a graph for each expression in terms of x.

Compare graphs to see which expression matches

Choose a convenient value for the y slider (for example, y = 1), making sure to avoid $x$ -values where the denominator $2x + 3y$ is zero (you will see breaks or asymptotes there). Compare the graphs: the option whose graph exactly overlaps the graph of (4x^2 - 9y^2)/(2x + 3y) for all allowed $x$ -values is the equivalent expression.

Step-by-step Explanation

Recognize a difference of squares

Look at the numerator $4x^2 - 9y^2$ .

Both terms are perfect squares:

$4x^2 = (2x)^2$
$9y^2 = (3y)^2$

So the numerator has the form $a^2 - b^2$ , which is a difference of squares.

Factor the numerator using the pattern

Use the identity $a^2 - b^2 = (a - b)(a + b)$ .

Here, $a = 2x$ and $b = 3y$ , so

4x^2 - 9y^2 = (2x - 3y)(2x + 3y)

Rewrite the fraction and simplify by canceling a common factor

Rewrite the original expression with the factored numerator:

\frac{4x^2 - 9y^2}{2x + 3y} = \frac{(2x - 3y)(2x + 3y)}{2x + 3y}

For all $x$ and $y$ such that $2x + 3y e 0$ , the factor $2x + 3y$ appears in both numerator and denominator, so it cancels:

\frac{(2x - 3y)(2x + 3y)}{2x + 3y} = 2x - 3y

Therefore, the rational expression is equivalent to $2x - 3y$ .

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation