Algebra Examples

$2 - 3 x < 5 (2 - x) \leq 3 (2 - x) + 10$

Step 1

Solve $2 - 3 x < 5 (2 - x)$ for $x$ .

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Step 1.1

Simplify $5 (2 - x)$ .

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Step 1.1.1

Rewrite.

$2 - 3 x < 0 + 0 + 5 (2 - x)$

Step 1.1.2

Simplify by adding zeros.

$2 - 3 x < 5 (2 - x)$

Step 1.1.3

Apply the distributive property.

$2 - 3 x < 5 \cdot 2 + 5 (- x)$

Step 1.1.4

Multiply.

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Step 1.1.4.1

Multiply $5$ by $2$ .

$2 - 3 x < 10 + 5 (- x)$

Step 1.1.4.2

Multiply $- 1$ by $5$ .

$2 - 3 x < 10 - 5 x$

Step 1.2

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

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Step 1.2.1

Add $5 x$ to both sides of the inequality.

$2 - 3 x + 5 x < 10$

Step 1.2.2

Add $- 3 x$ and $5 x$ .

$2 + 2 x < 10$

Step 1.3

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

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Step 1.3.1

Subtract $2$ from both sides of the inequality.

$2 x < 10 - 2$

Step 1.3.2

Subtract $2$ from $10$ .

$2 x < 8$

Step 1.4

Divide each term in $2 x < 8$ by $2$ and simplify.

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Step 1.4.1

Divide each term in $2 x < 8$ by $2$ .

$\frac{2 x}{2} < \frac{8}{2}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $2$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{8}{2}$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x < \frac{8}{2}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide $8$ by $2$ .

$x < 4$

Step 2

Solve $5 (2 - x) \leq 3 (2 - x) + 10$ for $x$ .

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Step 2.1

Simplify $5 (2 - x)$ .

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Step 2.1.1

Rewrite.

$0 + 0 + 5 (2 - x) \leq 3 (2 - x) + 10$

Step 2.1.2

Simplify by adding zeros.

$5 (2 - x) \leq 3 (2 - x) + 10$

Step 2.1.3

Apply the distributive property.

$5 \cdot 2 + 5 (- x) \leq 3 (2 - x) + 10$

Step 2.1.4

Multiply.

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Step 2.1.4.1

Multiply $5$ by $2$ .

$10 + 5 (- x) \leq 3 (2 - x) + 10$

Step 2.1.4.2

Multiply $- 1$ by $5$ .

$10 - 5 x \leq 3 (2 - x) + 10$

Step 2.2

Simplify $3 (2 - x) + 10$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$10 - 5 x \leq 3 \cdot 2 + 3 (- x) + 10$

Step 2.2.1.2

Multiply $3$ by $2$ .

$10 - 5 x \leq 6 + 3 (- x) + 10$

Step 2.2.1.3

Multiply $- 1$ by $3$ .

$10 - 5 x \leq 6 - 3 x + 10$

Step 2.2.2

Add $6$ and $10$ .

$10 - 5 x \leq - 3 x + 16$

Step 2.3

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

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Step 2.3.1

Add $3 x$ to both sides of the inequality.

$10 - 5 x + 3 x \leq 16$

Step 2.3.2

Add $- 5 x$ and $3 x$ .

$10 - 2 x \leq 16$

Step 2.4

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

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Step 2.4.1

Subtract $10$ from both sides of the inequality.

$- 2 x \leq 16 - 10$

Step 2.4.2

Subtract $10$ from $16$ .

$- 2 x \leq 6$

Step 2.5

Divide each term in $- 2 x \leq 6$ by $- 2$ and simplify.

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Step 2.5.1

Divide each term in $- 2 x \leq 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{6}{- 2}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $- 2$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{6}{- 2}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{6}{- 2}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $6$ by $- 2$ .

$x \geq - 3$

Step 3

Find the intersection of $x < 4$ and $x \geq - 3$ .

$- 3 \leq x < 4$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x < 4$

Interval Notation:

$[- 3, 4)$

Step 5