Finite Math Examples

$4 - x < 3 x + 7 < 20$

Step 1

Solve $4 - x < 3 x + 7$ for $x$ .

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Step 1.1

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

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Step 1.1.1

Subtract $3 x$ from both sides of the inequality.

$4 - x - 3 x < 7$

Step 1.1.2

Subtract $3 x$ from $- x$ .

$4 - 4 x < 7$

Step 1.2

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

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Step 1.2.1

Subtract $4$ from both sides of the inequality.

$- 4 x < 7 - 4$

Step 1.2.2

Subtract $4$ from $7$ .

$- 4 x < 3$

Step 1.3

Divide each term in $- 4 x < 3$ by $- 4$ and simplify.

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Step 1.3.1

Divide each term in $- 4 x < 3$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{3}{- 4}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{3}{- 4}$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x > \frac{3}{- 4}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Move the negative in front of the fraction.

$x > - \frac{3}{4}$

Step 2

Solve $3 x + 7 < 20$ 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

Subtract $7$ from both sides of the inequality.

$3 x < 20 - 7$

Step 2.1.2

Subtract $7$ from $20$ .

$3 x < 13$

Step 2.2

Divide each term in $3 x < 13$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x < 13$ by $3$ .

$\frac{3 x}{3} < \frac{13}{3}$

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.

$\frac{3 x}{3} < \frac{13}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{13}{3}$

Step 3

Find the intersection of $x > - \frac{3}{4}$ and $x < \frac{13}{3}$ .

$- \frac{3}{4} < x < \frac{13}{3}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- \frac{3}{4} < x < \frac{13}{3}$

Interval Notation:

$(- \frac{3}{4}, \frac{13}{3})$

Step 5