Finite Math Examples

- (2 (x - 1) - 7) \leq 9 x - (3 - x)

$- (2 (x - 1) - 7) \leq 9 x - (3 - x)$

Step 1

Rewrite so

x

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

9 x - (3 - x) \geq - (2 (x - 1) - 7)

$9 x - (3 - x) \geq - (2 (x - 1) - 7)$

Step 2

Simplify

9 x - (3 - x)

$9 x - (3 - x)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

9 x - 1 \cdot 3 - - x \geq - (2 (x - 1) - 7)

$9 x - 1 \cdot 3 - - x \geq - (2 (x - 1) - 7)$

Step 2.1.2

Multiply

- 1

$- 1$ by

3

$3$ .

9 x - 3 - - x \geq - (2 (x - 1) - 7)

$9 x - 3 - - x \geq - (2 (x - 1) - 7)$

Step 2.1.3

Multiply

- - x

$- - x$ .

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Step 2.1.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

9 x - 3 + 1 x \geq - (2 (x - 1) - 7)

$9 x - 3 + 1 x \geq - (2 (x - 1) - 7)$

Step 2.1.3.2

Multiply

x

$x$ by

1

$1$ .

9 x - 3 + x \geq - (2 (x - 1) - 7)

$9 x - 3 + x \geq - (2 (x - 1) - 7)$

9 x - 3 + x \geq - (2 (x - 1) - 7)

$9 x - 3 + x \geq - (2 (x - 1) - 7)$

9 x - 3 + x \geq - (2 (x - 1) - 7)

$9 x - 3 + x \geq - (2 (x - 1) - 7)$

Step 2.2

Add

9 x

$9 x$ and

x

$x$ .

10 x - 3 \geq - (2 (x - 1) - 7)

$10 x - 3 \geq - (2 (x - 1) - 7)$

10 x - 3 \geq - (2 (x - 1) - 7)

$10 x - 3 \geq - (2 (x - 1) - 7)$

Step 3

Simplify

- (2 (x - 1) - 7)

$- (2 (x - 1) - 7)$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

10 x - 3 \geq - (2 x + 2 \cdot - 1 - 7)

$10 x - 3 \geq - (2 x + 2 \cdot - 1 - 7)$

Step 3.1.2

Multiply

2

$2$ by

- 1

$- 1$ .

10 x - 3 \geq - (2 x - 2 - 7)

$10 x - 3 \geq - (2 x - 2 - 7)$

10 x - 3 \geq - (2 x - 2 - 7)

$10 x - 3 \geq - (2 x - 2 - 7)$

Step 3.2

Simplify by multiplying through.

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Step 3.2.1

Subtract

7

$7$ from

- 2

$- 2$ .

10 x - 3 \geq - (2 x - 9)

$10 x - 3 \geq - (2 x - 9)$

Step 3.2.2

Apply the distributive property.

10 x - 3 \geq - (2 x) - - 9

$10 x - 3 \geq - (2 x) - - 9$

Step 3.2.3

Multiply.

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Step 3.2.3.1

Multiply

2

$2$ by

- 1

$- 1$ .

10 x - 3 \geq - 2 x - - 9

$10 x - 3 \geq - 2 x - - 9$

Step 3.2.3.2

Multiply

- 1

$- 1$ by

- 9

$- 9$ .

10 x - 3 \geq - 2 x + 9

$10 x - 3 \geq - 2 x + 9$

10 x - 3 \geq - 2 x + 9

$10 x - 3 \geq - 2 x + 9$

10 x - 3 \geq - 2 x + 9

$10 x - 3 \geq - 2 x + 9$

10 x - 3 \geq - 2 x + 9

$10 x - 3 \geq - 2 x + 9$

Step 4

Move all terms containing

x

$x$ to the left side of the inequality.

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Step 4.1

Add

2 x

$2 x$ to both sides of the inequality.

10 x - 3 + 2 x \geq 9

$10 x - 3 + 2 x \geq 9$

Step 4.2

Add

10 x

$10 x$ and

2 x

$2 x$ .

12 x - 3 \geq 9

$12 x - 3 \geq 9$

12 x - 3 \geq 9

$12 x - 3 \geq 9$

Step 5

Move all terms not containing

x

$x$ to the right side of the inequality.

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Step 5.1

Add

3

$3$ to both sides of the inequality.

12 x \geq 9 + 3

$12 x \geq 9 + 3$

Step 5.2

Add

9

$9$ and

3

$3$ .

12 x \geq 12

$12 x \geq 12$

12 x \geq 12

$12 x \geq 12$

Step 6

Divide each term in

12 x \geq 12

$12 x \geq 12$ by

12

$12$ and simplify.

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Step 6.1

Divide each term in

12 x \geq 12

$12 x \geq 12$ by

12

$12$ .

\frac{12 x}{12} \geq \frac{12}{12}

$\frac{12 x}{12} \geq \frac{12}{12}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

12

$12$ .

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Step 6.2.1.1

Cancel the common factor.

\frac{12 x}{12} \geq \frac{12}{12}

$\frac{12 x}{12} \geq \frac{12}{12}$

Step 6.2.1.2

Divide

x

$x$ by

1

$1$ .

x \geq \frac{12}{12}

$x \geq \frac{12}{12}$

x \geq \frac{12}{12}

$x \geq \frac{12}{12}$

x \geq \frac{12}{12}

$x \geq \frac{12}{12}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide

12

$12$ by

12

$12$ .

x \geq 1

$x \geq 1$

x \geq 1

$x \geq 1$

x \geq 1

$x \geq 1$

Step 7

The result can be shown in multiple forms.

Inequality Form:

x \geq 1

$x \geq 1$

Interval Notation:

[1, \infty)

$[1, \infty)$