Pre-Algebra Examples

$x + 5 \leq 6 x + 20 \leq x + 80$

Step 1

Solve $x + 5 \leq 6 x + 20$ 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 $6 x$ from both sides of the inequality.

$x + 5 - 6 x \leq 20$

Step 1.1.2

Subtract $6 x$ from $x$ .

$- 5 x + 5 \leq 20$

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 $5$ from both sides of the inequality.

$- 5 x \leq 20 - 5$

Step 1.2.2

Subtract $5$ from $20$ .

$- 5 x \leq 15$

Step 1.3

Divide each term in $- 5 x \leq 15$ by $- 5$ and simplify.

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Step 1.3.1

Divide each term in $- 5 x \leq 15$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \geq \frac{15}{- 5}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 5$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} \geq \frac{15}{- 5}$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{15}{- 5}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $15$ by $- 5$ .

$x \geq - 3$

Step 2

Solve $6 x + 20 \leq x + 80$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Subtract $x$ from both sides of the inequality.

$6 x + 20 - x \leq 80$

Step 2.1.2

Subtract $x$ from $6 x$ .

$5 x + 20 \leq 80$

Step 2.2

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

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Step 2.2.1

Subtract $20$ from both sides of the inequality.

$5 x \leq 80 - 20$

Step 2.2.2

Subtract $20$ from $80$ .

$5 x \leq 60$

Step 2.3

Divide each term in $5 x \leq 60$ by $5$ and simplify.

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Step 2.3.1

Divide each term in $5 x \leq 60$ by $5$ .

$\frac{5 x}{5} \leq \frac{60}{5}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \leq \frac{60}{5}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{60}{5}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $60$ by $5$ .

$x \leq 12$

Step 3

Find the intersection of $x \geq - 3$ and $x \leq 12$ .

$- 3 \leq x \leq 12$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x \leq 12$

Interval Notation:

$[- 3, 12]$

Step 5