Algebra Examples

$- x + 10 \leq 4$ and $- 1 \leq - x + 10$

Step 1

Simplify the first inequality.

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Step 1.1

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

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Step 1.1.1

Subtract $10$ from both sides of the inequality.

$- x \leq 4 - 10$

Step 1.1.2

Subtract $10$ from $4$ .

$- x \leq - 6$

Step 1.2

Divide each term in $- x \leq - 6$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- x \leq - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{- 6}{- 1}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{- 6}{- 1}$

Step 1.2.2.2

Divide $x$ by $1$ .

$x \geq \frac{- 6}{- 1}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $- 6$ by $- 1$ .

$x \geq 6$

Step 2

Simplify the second inequality.

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Step 2.1

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

$- x + 10 \geq - 1$

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 $10$ from both sides of the inequality.

$- x \geq - 1 - 10$

Step 2.2.2

Subtract $10$ from $- 1$ .

$- x \geq - 11$

Step 2.3

Divide each term in $- x \geq - 11$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- x \geq - 11$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 11}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 11}{- 1}$

Step 2.3.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 11}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 11$ by $- 1$ .

$x \leq 11$

Step 3

The intersection consists of the elements that are contained in both intervals.

$6 \leq x \leq 11$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$6 \leq x \leq 11$

Interval Notation:

$[6, 11]$

Step 5