Precalculus Examples

$5 < 3 x - 1 < 14$

Step 1

Solve $5 < 3 x - 1$ for $x$ .

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Step 1.1

Rewrite so $x$ is on the left side of the inequality.

$3 x - 1 > 5$

Step 1.2

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

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Step 1.2.1

Add $1$ to both sides of the inequality.

$3 x > 5 + 1$

Step 1.2.2

Add $5$ and $1$ .

$3 x > 6$

Step 1.3

Divide each term in $3 x > 6$ by $3$ and simplify.

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Step 1.3.1

Divide each term in $3 x > 6$ by $3$ .

$\frac{3 x}{3} > \frac{6}{3}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $3$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} > \frac{6}{3}$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x > \frac{6}{3}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $6$ by $3$ .

$x > 2$

Step 2

Solve $3 x - 1 < 14$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Add $1$ to both sides of the inequality.

$3 x < 14 + 1$

Step 2.1.2

Add $14$ and $1$ .

$3 x < 15$

Step 2.2

Divide each term in $3 x < 15$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x < 15$ by $3$ .

$\frac{3 x}{3} < \frac{15}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{15}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{15}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $15$ by $3$ .

$x < 5$

Step 3

Find the intersection of $x > 2$ and $x < 5$ .

$2 < x < 5$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 5$

Interval Notation:

$(2, 5)$

Step 5