Precalculus Examples

$6 - \frac{1}{2} x = 3 x + 1 + g (x)$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 x + 1 + g x = 6 - \frac{1}{2} x$

Step 2

Combine $x$ and $\frac{1}{2}$ .

$3 x + 1 + g x = 6 - \frac{x}{2}$

Step 3

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 3.1

Add $\frac{x}{2}$ to both sides of the equation.

$3 x + 1 + g x + \frac{x}{2} = 6$

Step 3.2

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1 + g x + 3 x \cdot \frac{2}{2} + \frac{x}{2} = 6$

Step 3.3

Combine $3 x$ and $\frac{2}{2}$ .

$1 + g x + \frac{3 x \cdot 2}{2} + \frac{x}{2} = 6$

Step 3.4

Combine the numerators over the common denominator.

$1 + g x + \frac{3 x \cdot 2 + x}{2} = 6$

Step 3.5

Simplify each term.

Tap for more steps...

Step 3.5.1

Simplify the numerator.

Tap for more steps...

Step 3.5.1.1

Factor $x$ out of $3 x \cdot 2 + x$ .

Tap for more steps...

Step 3.5.1.1.1

Factor $x$ out of $3 x \cdot 2$ .

$1 + g x + \frac{x (3 \cdot 2) + x}{2} = 6$

Step 3.5.1.1.2

Raise $x$ to the power of $1$ .

$1 + g x + \frac{x (3 \cdot 2) + x^{1}}{2} = 6$

Step 3.5.1.1.3

Factor $x$ out of $x^{1}$ .

$1 + g x + \frac{x (3 \cdot 2) + x \cdot 1}{2} = 6$

Step 3.5.1.1.4

Factor $x$ out of $x (3 \cdot 2) + x \cdot 1$ .

$1 + g x + \frac{x (3 \cdot 2 + 1)}{2} = 6$

Step 3.5.1.2

Multiply $3$ by $2$ .

$1 + g x + \frac{x (6 + 1)}{2} = 6$

Step 3.5.1.3

Add $6$ and $1$ .

$1 + g x + \frac{x \cdot 7}{2} = 6$

Step 3.5.2

Move $7$ to the left of $x$ .

$1 + g x + \frac{7 x}{2} = 6$

Step 4

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 4.1

Subtract $1$ from both sides of the equation.

$g x + \frac{7 x}{2} = 6 - 1$

Step 4.2

Subtract $1$ from $6$ .

$g x + \frac{7 x}{2} = 5$

Step 5

Factor $x$ out of $g x + \frac{7 x}{2}$ .

Tap for more steps...

Step 5.1

Factor $x$ out of $g x$ .

$x g + \frac{7 x}{2} = 5$

Step 5.2

Factor $x$ out of $\frac{7 x}{2}$ .

$x g + x (\frac{7}{2}) = 5$

Step 5.3

Factor $x$ out of $x g + x \frac{7}{2}$ .

$x (g + \frac{7}{2}) = 5$

Step 6

Divide each term in $x (g + \frac{7}{2}) = 5$ by $g + \frac{7}{2}$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $x (g + \frac{7}{2}) = 5$ by $g + \frac{7}{2}$ .

$\frac{x (g + \frac{7}{2})}{g + \frac{7}{2}} = \frac{5}{g + \frac{7}{2}}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $g + \frac{7}{2}$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{x (g + \frac{7}{2})}{g + \frac{7}{2}} = \frac{5}{g + \frac{7}{2}}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{g + \frac{7}{2}}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Simplify the denominator.

Tap for more steps...

Step 6.3.1.1

To write $g$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{5}{g \cdot \frac{2}{2} + \frac{7}{2}}$

Step 6.3.1.2

Combine $g$ and $\frac{2}{2}$ .

$x = \frac{5}{\frac{g \cdot 2}{2} + \frac{7}{2}}$

Step 6.3.1.3

Combine the numerators over the common denominator.

$x = \frac{5}{\frac{g \cdot 2 + 7}{2}}$

Step 6.3.1.4

Move $2$ to the left of $g$ .

$x = \frac{5}{\frac{2 g + 7}{2}}$

Step 6.3.2

Multiply the numerator by the reciprocal of the denominator.

$x = 5 \frac{2}{2 g + 7}$

Step 6.3.3

Multiply $5 \frac{2}{2 g + 7}$ .

Tap for more steps...

Step 6.3.3.1

Combine $5$ and $\frac{2}{2 g + 7}$ .

$x = \frac{5 \cdot 2}{2 g + 7}$

Step 6.3.3.2

Multiply $5$ by $2$ .

$x = \frac{10}{2 g + 7}$

Step 7

To rewrite as a function of $g$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $g$ is on the other side.

$f (g) = \frac{10}{2 g + 7}$