Algebra Examples

$\frac{3}{4} x + 7 < 1$ or $9 (4 - x) \leq 36$

Step 1

Simplify the first inequality.

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Step 1.1

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

$\frac{3 x}{4} + 7 < 1$ or $9 (4 - x) \leq 36$

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.

$\frac{3 x}{4} < 1 - 7$ or $9 (4 - x) \leq 36$

Step 1.2.2

Subtract $7$ from $1$ .

$\frac{3 x}{4} < - 6$ or $9 (4 - x) \leq 36$

Step 1.3

Multiply both sides by $4$ .

$\frac{3 x}{4} \cdot 4 < - 6 \cdot 4$ or $9 (4 - x) \leq 36$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Cancel the common factor of $4$ .

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Step 1.4.1.1.1

Cancel the common factor.

$\frac{3 x}{4} \cdot 4 < - 6 \cdot 4$ or $9 (4 - x) \leq 36$

Step 1.4.1.1.2

Rewrite the expression.

$3 x < - 6 \cdot 4$ or $9 (4 - x) \leq 36$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Multiply $- 6$ by $4$ .

$3 x < - 24$ or $9 (4 - x) \leq 36$

Step 1.5

Divide each term in $3 x < - 24$ by $3$ and simplify.

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Step 1.5.1

Divide each term in $3 x < - 24$ by $3$ .

$\frac{3 x}{3} < \frac{- 24}{3}$ or $9 (4 - x) \leq 36$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 24}{3}$ or $9 (4 - x) \leq 36$

Step 1.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 24}{3}$ or $9 (4 - x) \leq 36$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $- 24$ by $3$ .

$x < - 8$ or $9 (4 - x) \leq 36$

Step 2

Simplify the second inequality.

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Step 2.1

Divide each term in $9 (4 - x) \leq 36$ by $9$ and simplify.

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Step 2.1.1

Divide each term in $9 (4 - x) \leq 36$ by $9$ .

$x < - 8$ or $\frac{9 (4 - x)}{9} \leq \frac{36}{9}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $9$ .

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Step 2.1.2.1.1

Cancel the common factor.

$x < - 8$ or $\frac{9 (4 - x)}{9} \leq \frac{36}{9}$

Step 2.1.2.1.2

Divide $4 - x$ by $1$ .

$x < - 8$ or $4 - x \leq \frac{36}{9}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $36$ by $9$ .

$x < - 8$ or $4 - x \leq 4$

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 $4$ from both sides of the inequality.

$x < - 8$ or $- x \leq 4 - 4$

Step 2.2.2

Subtract $4$ from $4$ .

$x < - 8$ or $- x \leq 0$

Step 2.3

Divide each term in $- x \leq 0$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- x \leq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$x < - 8$ or $\frac{- x}{- 1} \geq \frac{0}{- 1}$

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.

$x < - 8$ or $\frac{x}{1} \geq \frac{0}{- 1}$

Step 2.3.2.2

Divide $x$ by $1$ .

$x < - 8$ or $x \geq \frac{0}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $- 1$ .

$x < - 8$ or $x \geq 0$

Step 3

Convert the inequality to interval notation.

$(- \infty, - 8) \cup [0, \infty)$

Step 4