Algebra Examples

$7 - \frac{4}{3} x < - 5$ and $\frac{1}{2} x - 7 < - 6$

Step 1

Simplify the first inequality.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x$ and $\frac{4}{3}$ .

$7 - \frac{x \cdot 4}{3} < - 5$ and $\frac{1}{2} x - 7 < - 6$

Step 1.1.2

Move $4$ to the left of $x$ .

$7 - \frac{4 x}{3} < - 5$ and $\frac{1}{2} x - 7 < - 6$

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 $7$ from both sides of the inequality.

$- \frac{4 x}{3} < - 5 - 7$ and $\frac{1}{2} x - 7 < - 6$

Step 1.2.2

Subtract $7$ from $- 5$ .

$- \frac{4 x}{3} < - 12$ and $\frac{1}{2} x - 7 < - 6$

Step 1.3

Divide each term in $- \frac{4 x}{3} < - 12$ by $- 1$ and simplify.

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Step 1.3.1

Divide each term in $- \frac{4 x}{3} < - 12$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{4 x}{3}}{- 1} > \frac{- 12}{- 1}$ and $\frac{1}{2} x - 7 < - 6$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{4 x}{3}}{1} > \frac{- 12}{- 1}$ and $\frac{1}{2} x - 7 < - 6$

Step 1.3.2.2

Divide $\frac{4 x}{3}$ by $1$ .

$\frac{4 x}{3} > \frac{- 12}{- 1}$ and $\frac{1}{2} x - 7 < - 6$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $- 12$ by $- 1$ .

$\frac{4 x}{3} > 12$ and $\frac{1}{2} x - 7 < - 6$

Step 1.4

Multiply both sides by $3$ .

$\frac{4 x}{3} \cdot 3 > 12 \cdot 3$ and $\frac{1}{2} x - 7 < - 6$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

Cancel the common factor of $3$ .

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Step 1.5.1.1.1

Cancel the common factor.

$\frac{4 x}{3} \cdot 3 > 12 \cdot 3$ and $\frac{1}{2} x - 7 < - 6$

Step 1.5.1.1.2

Rewrite the expression.

$4 x > 12 \cdot 3$ and $\frac{1}{2} x - 7 < - 6$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Multiply $12$ by $3$ .

$4 x > 36$ and $\frac{1}{2} x - 7 < - 6$

Step 1.6

Divide each term in $4 x > 36$ by $4$ and simplify.

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Step 1.6.1

Divide each term in $4 x > 36$ by $4$ .

$\frac{4 x}{4} > \frac{36}{4}$ and $\frac{1}{2} x - 7 < - 6$

Step 1.6.2

Simplify the left side.

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Step 1.6.2.1

Cancel the common factor of $4$ .

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Step 1.6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} > \frac{36}{4}$ and $\frac{1}{2} x - 7 < - 6$

Step 1.6.2.1.2

Divide $x$ by $1$ .

$x > \frac{36}{4}$ and $\frac{1}{2} x - 7 < - 6$

Step 1.6.3

Simplify the right side.

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Step 1.6.3.1

Divide $36$ by $4$ .

$x > 9$ and $\frac{1}{2} x - 7 < - 6$

Step 2

Simplify the second inequality.

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Step 2.1

Combine $\frac{1}{2}$ and $x$ .

$x > 9$ and $\frac{x}{2} - 7 < - 6$

Step 2.2

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

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Step 2.2.1

Add $7$ to both sides of the inequality.

$x > 9$ and $\frac{x}{2} < - 6 + 7$

Step 2.2.2

Add $- 6$ and $7$ .

$x > 9$ and $\frac{x}{2} < 1$

Step 2.3

Multiply both sides by $2$ .

$x > 9$ and $\frac{x}{2} \cdot 2 < 1 \cdot 2$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1

Cancel the common factor.

$x > 9$ and $\frac{x}{2} \cdot 2 < 1 \cdot 2$

Step 2.4.1.1.2

Rewrite the expression.

$x > 9$ and $x < 1 \cdot 2$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $2$ by $1$ .

$x > 9$ and $x < 2$

Step 3

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

No solution