Question

Solve for xx. Express your answer as a proper or improper fraction in simplest terms.
minus, one half, equals, start fraction, 4, divided by, 9, end fraction, x, plus, start fraction, 5, divided by, 6, end fraction
−
2
1
​
=
9
4
​
x+
6
5
​

Answer 1

To solve the equation

\[
-\frac{1}{2} = \frac{4}{9}x + \frac{5}{6}
\]

we will first isolate \( x \).

1. **Eliminate the constant term \( \frac{5}{6} \) from the right side by subtracting it from both sides**:

\[
-\frac{1}{2} - \frac{5}{6} = \frac{4}{9}x
\]

To combine the left-hand side, we need a common denominator. The least common multiple of 2 and 6 is 6.

Convert \( -\frac{1}{2} \) to sixths:

\[
-\frac{1}{2} = -\frac{3}{6}
\]

Now, substituting this into the equation gives:

\[
-\frac{3}{6} - \frac{5}{6} = \frac{4}{9}x
\]

Combine the fractions:

\[
-\frac{3 + 5}{6} = \frac{4}{9}x
\]

This simplifies to:

\[
-\frac{8}{6} = \frac{4}{9}x
\]

2. **Simplify \( -\frac{8}{6} \)**:

\[
-\frac{8}{6} = -\frac{4}{3}
\]

So now we have:

\[
-\frac{4}{3} = \frac{4}{9}x
\]

3. **Multiply both sides by the reciprocal of \( \frac{4}{9} \) to solve for \( x \)**. The reciprocal is \( \frac{9}{4} \):

\[
x = -\frac{4}{3} \cdot \frac{9}{4}
\]

The \( 4 \)s cancel:

\[
x = -\frac{9}{3}
\]

This simplifies to:

\[
x = -3
\]

Thus, the solution is:

\[
\boxed{-3}
\]