Algebra Examples

$3 x + 8 + 2 a x \geq 3 a x - 4 a$

Step 1

Rewrite so $a$ is on the left side of the inequality.

$3 a x - 4 a \leq 3 x + 8 + 2 a x$

Step 2

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

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Step 2.1

Subtract $2 a x$ from both sides of the inequality.

$3 a x - 4 a - 2 a x \leq 3 x + 8$

Step 2.2

Subtract $2 a x$ from $3 a x$ .

$- 4 a + 1 a x \leq 3 x + 8$

Step 2.3

Multiply $a$ by $1$ .

$- 4 a + a x \leq 3 x + 8$

Step 3

Factor $a$ out of $- 4 a + a x$ .

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Step 3.1

Factor $a$ out of $- 4 a$ .

$a \cdot - 4 + a x \leq 3 x + 8$

Step 3.2

Factor $a$ out of $a x$ .

$a \cdot - 4 + a (x) \leq 3 x + 8$

Step 3.3

Factor $a$ out of $a \cdot - 4 + a (x)$ .

$a (- 4 + x) \leq 3 x + 8$

Step 4

Divide each term in $a (- 4 + x) \leq 3 x + 8$ by $- 4 + x$ and simplify.

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Step 4.1

Divide each term in $a (- 4 + x) \leq 3 x + 8$ by $- 4 + x$ .

$\frac{a (- 4 + x)}{- 4 + x} \leq \frac{3 x}{- 4 + x} + \frac{8}{- 4 + x}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4 + x$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{a (- 4 + x)}{- 4 + x} \leq \frac{3 x}{- 4 + x} + \frac{8}{- 4 + x}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a \leq \frac{3 x}{- 4 + x} + \frac{8}{- 4 + x}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$a \leq \frac{3 x + 8}{- 4 + x}$

Step 4.3.2

Rewrite $- 4$ as $- 1 (4)$ .

$a \leq \frac{3 x + 8}{- 1 (4) + x}$

Step 4.3.3

Factor $- 1$ out of $x$ .

$a \leq \frac{3 x + 8}{- 1 (4) - 1 (- x)}$

Step 4.3.4

Factor $- 1$ out of $- 1 (4) - 1 (- x)$ .

$a \leq \frac{3 x + 8}{- 1 (4 - x)}$

Step 4.3.5

Move the negative in front of the fraction.

$a \leq - \frac{3 x + 8}{4 - x}$