After distributing, be sure to distribute the minus sign in front of $\dfrac{5}{6}(x + 7)$, then combine all the $x$ terms together and all the constant terms together on the left side.

Once you have an equation with fractions, multiply every term on both sides by a common multiple of the denominators (12 works here) to turn it into an equation with whole numbers.

Solve the resulting simple equation for $x$, then write your final answer as a decimal so you can see which two consecutive integers it falls between.

In Desmos, type y = (3/4)(2x - 5) - (5/6)(x + 7) on one line and y = x/2 + 1 on another line so you have two graphs on the same coordinate plane.

Zoom and pan until you can see where the two graphs intersect. Tap or click on the intersection point and note the x-coordinate; that value of $x$ solves the equation.

Look at the x-coordinate from Desmos and determine which pair of consecutive integers it lies between, then select the answer choice whose interval matches that pair.

Start by expanding the fractions:

• Left side:
  • $\dfrac{3}{4}(2x - 5) = \dfrac{3}{4}\cdot 2x - \dfrac{3}{4}\cdot 5 = \dfrac{3}{2}x - \dfrac{15}{4}$
  • $\dfrac{5}{6}(x + 7) = \dfrac{5}{6}x + \dfrac{35}{6}$

So the equation becomes:

Distribute the minus sign across the parentheses:

Now combine like terms on the left:

• For $x$ terms: $\dfrac{3}{2} - \dfrac{5}{6} = \dfrac{9}{6} - \dfrac{5}{6} = \dfrac{4}{6} = \dfrac{2}{3}$, so the $x$ term is $\dfrac{2}{3}x$.
• For constants: $-\dfrac{15}{4} - \dfrac{35}{6} = -\dfrac{45}{12} - \dfrac{70}{12} = -\dfrac{115}{12}$.

So the equation simplifies to:

To remove the fractions, multiply every term on both sides by 12 (the least common multiple of 3, 12, and 2):

Compute each term:

• $12\cdot \dfrac{2}{3}x = 8x$
• $12\cdot \dfrac{115}{12} = 115$
• $12\cdot \dfrac{x}{2} = 6x$
• $12\cdot 1 = 12$

So the equation becomes:

Now solve $8x - 115 = 6x + 12$.

1. Subtract $6x$ from both sides:

which simplifies to

2. Add 115 to both sides:

Now you have a simple equation with no fractions.

From $2x = 127$, divide both sides by 2:

Convert this to a decimal to see which integers it is between:

The number $63.5$ lies between 63 and 64, so the solution $x$ is between 63 and 64.