Which of the following are solutions to the given equation, where is a constant and ? I. II. III.

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 111 | Free SAT Prep | Aniko

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Question 111·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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\frac{x - 4}{x - b} = x - 4

Which of the following are solutions to the given equation, where $b$ is a constant and $b < 3$ ?

I. $b$

II. $4$

III. $b + 1$

For rational equations with a variable in the denominator, first identify and exclude any values that make the denominator zero, since those can never be solutions. Then, clear the fraction by multiplying both sides by the nonzero denominator, rearrange the resulting equation to equal 0, and factor to find candidate solutions. Finally, check each candidate against the domain restriction (no zero denominators) and against the original equation, and then match your valid solutions with the answer choices.

Hints

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Check the domain first

Look at the denominator $x-b$ . For which value of $x$ is the expression $\dfrac{x-4}{x-b}$ not defined?

Eliminate the fraction

Assuming $x \ne b$ , what can you multiply both sides of the equation by so that the fraction disappears? After that, rearrange the equation so everything is on one side.

Factor and solve

Once you have an equation equal to 0, try factoring. What values of $x$ make each factor zero, and do any of these conflict with the restriction from the denominator?

Compare with I, II, III

Take the $x$ -values you found and see which of I ( $b$ ), II ( $4$ ), and III ( $b+1$ ) match them, remembering that $x=b$ makes the original equation undefined.

Desmos Guide

Set up the equation with a parameter

In Desmos, type b=2 (or any number less than 3 to respect $b<3$ ). Then enter two functions: f(x)=(x-4)/(x-b) and g(x)=x-4.

See where the graphs intersect

Look at the graph to find the $x$ -values where the curves of $f(x)$ and $g(x)$ intersect. These $x$ -coordinates are the solutions to the equation for that chosen value of $b$ ; note their relationship to 4 and $b$ .

Test the candidate values from I, II, and III

Create a new expression like (x-4)/(x-b) - (x-4) and then, in separate lines, substitute each candidate: replace x with b, then with 4, then with b+1 (using the same numeric value of b from step 1). Check which substitutions make the expression undefined (division by zero) and which make it equal to 0; those are the values that satisfy the equation.

Step-by-step Explanation

Determine when the equation is defined

The equation is

\frac{x-4}{x-b} = x-4.

Because there is a denominator $x-b$ , division by zero is not allowed, so

x \ne b.

This means $x=b$ cannot be a solution under any circumstances.

Clear the fraction carefully

Now solve the equation for $x$ , assuming $x \ne b$ so that $x-b \ne 0$ . Multiply both sides by $x-b$ :

\frac{x-4}{x-b}(x-b) = (x-4)(x-b).

This simplifies to

\begin{aligned} x-4 &= (x-4)(x-b) \\ 0 &= (x-4)(x-b) - (x-4) \\ 0 &= (x-4)\big[(x-b)-1\big] \\ 0 &= (x-4)(x-b-1). \end{aligned}

So the equation becomes a product of two factors equal to zero.

Find all algebraic solutions

From

(x-4)(x-b-1) = 0,

we get two possible solutions by setting each factor equal to zero:

$x-4 = 0$ gives $x = 4$ .
$x-b-1 = 0$ gives $x = b+1$ .

So the candidate solutions are $x=4$ and $x=b+1$ (remember we already know $x \ne b$ ).

Match candidates with I, II, III and apply restrictions

Now compare with the three listed values:

I. $b$ : This is not allowed because the original equation is undefined at $x=b$ due to division by zero.
II. $4$ : This matches one of our algebraic solutions. Also, since $b<3$ , we know $b \ne 4$ , so the denominator $4-b$ is not zero, and $x=4$ really works.
III. $b+1$ : This matches the other algebraic solution. At $x=b+1$ , the denominator is $(b+1)-b = 1$ , which is not zero, so $x=b+1$ also works.

Therefore, the solutions among the choices are II and III only.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation