Question 537·Hard·Math

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Solve for $x$ .

\frac{5}{3}\left(2-\frac{3x-4}{2}\right)+\frac{1}{4}(x+6)=7-\frac{5x}{6}

Fractions and parentheses on both sides point to a linear equation: the unknown appears only to the first power. On a timed test, use a one-variable regression in Desmos to solve without clearing every fraction, then convert its decimal result to an exact fraction. Keep the entire numerator grouped when you enter a fraction. Dropping the minus before a grouped numerator changes the equation.

Hints

Hint 1
For a one-variable regression, replace every $x$ with $x_1$ and the equals sign with $\sim$ . Desmos then finds the value of $x_1$ that makes the two sides equal.
Hint 2
The minus before $\frac{3x-4}{2}$ subtracts the whole fraction, not only $3x$ . Keep $3x-4$ together under the fraction bar when you type the equation.
Hint 3
A regression reports a decimal value under PARAMETERS. To see whether it has an exact fraction form, type $x_1$ on a new line and use the fraction button beside that line's result.

Step-by-step

Approach 1: Solve the equation in Desmos
Step 1
Find the value that makes both sides equal
Type the equation below, replacing every $x$ with $x_1$ and $=$ with $\sim$ :
$\frac{5}{3}\left(2-\frac{3x_1-4}{2}\right)+\frac{1}{4}(x_1+6)\sim7-\frac{5x_1}{6}$
Desmos treats $x_1$ as a number to find. Keep $3x_1-4$ together under its fraction bar. Under PARAMETERS, Desmos shows $x_1\approx0.823529$ .
Step 2
Read the exact fraction
Type $x_1$ on the next line. Desmos displays its decimal value with a fraction button; tap it to see $\frac{14}{17}$ . That is the exact value of $x$ that solves the equation. Choice D.

Approach 2: Clear the fractions by hand
Step 1
Remove the outer denominators
The denominators $3$ , $2$ , $4$ , and $6$ all divide $12$ . Multiply every term on both sides by $12$ to keep the equation equal:
$20\left(2-\frac{3x-4}{2}\right)+3(x+6)=84-10x$
Step 2
Distribute the 20
Multiply the $20$ by both terms in its parentheses; in the second term, $20$ divided by $2$ is $10$ :
$40-10(3x-4)+3(x+6)=84-10x$
Step 3
Distribute the negative 10
The negative sign reaches both terms of $3x-4$ . Distribute $-10$ :
$40-30x+40+3(x+6)=84-10x$
Step 4
Distribute the 3
Multiply $3$ by each term in $x+6$ :
$40-30x+40+3x+18=84-10x$
Step 5
Combine like terms
Like terms have the same variable part. Combine the $x$ -terms and the plain numbers on the left:
$98-27x=84-10x$
Step 6
Gather the x-terms
Add $10x$ to both sides so the $x$ -terms are together:
$98-17x=84$
Step 7
Move the constant
Subtract $98$ from both sides; $84-98$ is negative:
$-17x=-14$
Step 8
Divide by the coefficient
Divide both sides by $-17$ to isolate $x$ :
$x=\frac{-14}{-17}$
Step 9
Simplify the signs
A negative divided by a negative is positive, so $x=\frac{14}{17}$ . Choice D.

Question 537·Hard·Math

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