Question 65·200 Super-Hard SAT Math Questions·Algebra

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\frac{x-1}{5}-\frac{x+2}{6}+\frac{2x-3}{8}=\frac{x}{10},

the value of $2x-10$ is between which of the following pairs of values?

A linear equation has its unknown only to the first power. Several fractions are a cue to solve by Desmos regression: replace $x$ with $x_1$ and $=$ with $\sim$ . Then calculate the expression being compared and find which endpoints surround it. Don’t choose the pair containing $x$ , and don’t round a value near $0$ to $0$ .

Hints

Hint 1
The fractions make this equation slower to solve by hand, but it still has one unknown. A Desmos regression can find its value. To make Desmos solve rather than graph, change every $x$ to $x_1$ and the equals sign to $\sim$ .
Hint 2
Solving for $x$ is only the first job. The question asks where a different number falls. Once Desmos reports $x_1$ , what could you type on the next line to calculate $2x_1-10$ ?

Step-by-step

Solve and compare in Desmos
Step 1
Find the value of x
The given equation has fractions but only one unknown, so use a regression to find it without clearing denominators. Type the equation with every $x$ changed to $x_1$ and $=$ changed to $\sim$ . That tells Desmos to solve for $x_1$ instead of graphing $x$ . Under PARAMETERS, Desmos reports $x_1\approx4.954545$ , so $x\approx4.954545$ .
Step 2
Calculate what the question asks for
Keep the equation and type $2x_1-10$ on the next line. Desmos prints $-0.090909\ldots$ . That value is less than $0$ but greater than $-1$ : $-1<-0.090909\ldots<0$ . Compare the requested expression, not just the value of $x$ . So $2x-10$ is between $-1$ and $0$ . Choice B.

Question 65·200 Super-Hard SAT Math Questions·Algebra

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