Algebra Examples

$7 x - 8 > 9 (x - 1)$ and $3 x + 2 < 5 x + 3$

Step 1

Simplify the first inequality.

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Step 1.1

Simplify $9 (x - 1)$ .

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Step 1.1.1

Rewrite.

$7 x - 8 > 0 + 0 + 9 (x - 1)$ and $3 x + 2 < 5 x + 3$

Step 1.1.2

Simplify by adding zeros.

$7 x - 8 > 9 (x - 1)$ and $3 x + 2 < 5 x + 3$

Step 1.1.3

Apply the distributive property.

$7 x - 8 > 9 x + 9 \cdot - 1$ and $3 x + 2 < 5 x + 3$

Step 1.1.4

Multiply $9$ by $- 1$ .

$7 x - 8 > 9 x - 9$ and $3 x + 2 < 5 x + 3$

Step 1.2

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

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Step 1.2.1

Subtract $9 x$ from both sides of the inequality.

$7 x - 8 - 9 x > - 9$ and $3 x + 2 < 5 x + 3$

Step 1.2.2

Subtract $9 x$ from $7 x$ .

$- 2 x - 8 > - 9$ and $3 x + 2 < 5 x + 3$

Step 1.3

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

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Step 1.3.1

Add $8$ to both sides of the inequality.

$- 2 x > - 9 + 8$ and $3 x + 2 < 5 x + 3$

Step 1.3.2

Add $- 9$ and $8$ .

$- 2 x > - 1$ and $3 x + 2 < 5 x + 3$

Step 1.4

Divide each term in $- 2 x > - 1$ by $- 2$ and simplify.

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Step 1.4.1

Divide each term in $- 2 x > - 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{- 1}{- 2}$ and $3 x + 2 < 5 x + 3$

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.

$\frac{- 2 x}{- 2} < \frac{- 1}{- 2}$ and $3 x + 2 < 5 x + 3$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 1}{- 2}$ and $3 x + 2 < 5 x + 3$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Dividing two negative values results in a positive value.

$x < \frac{1}{2}$ and $3 x + 2 < 5 x + 3$

Step 2

Simplify the second inequality.

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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 $5 x$ from both sides of the inequality.

$x < \frac{1}{2}$ and $3 x + 2 - 5 x < 3$

Step 2.1.2

Subtract $5 x$ from $3 x$ .

$x < \frac{1}{2}$ and $- 2 x + 2 < 3$

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 $2$ from both sides of the inequality.

$x < \frac{1}{2}$ and $- 2 x < 3 - 2$

Step 2.2.2

Subtract $2$ from $3$ .

$x < \frac{1}{2}$ and $- 2 x < 1$

Step 2.3

Divide each term in $- 2 x < 1$ by $- 2$ and simplify.

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Step 2.3.1

Divide each term in $- 2 x < 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$x < \frac{1}{2}$ and $\frac{- 2 x}{- 2} > \frac{1}{- 2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$x < \frac{1}{2}$ and $\frac{- 2 x}{- 2} > \frac{1}{- 2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{1}{2}$ and $x > \frac{1}{- 2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$x < \frac{1}{2}$ and $x > - \frac{1}{2}$

Step 3

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

$- \frac{1}{2} < x < \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- \frac{1}{2} < x < \frac{1}{2}$

Interval Notation:

$(- \frac{1}{2}, \frac{1}{2})$

Step 5