Algebra Examples

${(6 x + 3)}^{- 1} < 0$

Step 1

Simplify ${(6 x + 3)}^{- 1}$ .

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{6 x + 3} < 0$

Step 1.2

Factor $3$ out of $6 x + 3$ .

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Step 1.2.1

Factor $3$ out of $6 x$ .

$\frac{1}{3 (2 x) + 3} < 0$

Step 1.2.2

Factor $3$ out of $3$ .

$\frac{1}{3 (2 x) + 3 (1)} < 0$

Step 1.2.3

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

$\frac{1}{3 (2 x + 1)} < 0$

Step 2

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$2 x + 1 = 0$

Step 3

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 4

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 5

The solution consists of all of the true intervals.

$x < - \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{1}{2}$

Interval Notation:

$(- \infty, - \frac{1}{2})$

Step 7