Algebra Examples

$\frac{a}{- 2} < - 1$ or $- 4 a + 3 \geq 23$

Step 1

Simplify the first inequality.

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Step 1.1

Multiply both sides by $- 2$ .

$\frac{a}{- 2} \cdot - 2 > 2$ or $- 4 a + 3 \geq 23$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Simplify $\frac{a}{- 2} \cdot - 2$ .

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Step 1.2.1.1.1

Cancel the common factor of $2$ .

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Step 1.2.1.1.1.1

Factor $2$ out of $- 2$ .

$\frac{a}{2 (- 1)} \cdot - 2 > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.1.2

Factor $2$ out of $- 2$ .

$\frac{a}{2 \cdot - 1} \cdot (2 \cdot - 1) > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.1.3

Cancel the common factor.

$\frac{a}{2 \cdot - 1} \cdot (2 \cdot - 1) > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.1.4

Rewrite the expression.

$\frac{a}{- 1} \cdot - 1 > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.2

Combine $\frac{a}{- 1}$ and $- 1$ .

$\frac{a \cdot - 1}{- 1} > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.3

Simplify the expression.

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Step 1.2.1.1.3.1

Move the negative one from the denominator of $\frac{a \cdot - 1}{- 1}$ .

$- 1 \cdot (a \cdot - 1) > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.3.2

Rewrite $- 1 \cdot (a \cdot - 1)$ as $- (a \cdot - 1)$ .

$- (a \cdot - 1) > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.3.3

Move $- 1$ to the left of $a$ .

$- (- 1 \cdot a) > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.3.4

Rewrite $- 1 a$ as $- a$ .

$a > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.4

Multiply $- - a$ .

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Step 1.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 a > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.1.1.4.2

Multiply $a$ by $1$ .

$a > 2$ or $- 4 a + 3 \geq 23$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Multiply $- 1$ by $- 2$ .

$a > 2$ or $- 4 a + 3 \geq 23$

Step 2

Simplify the second inequality.

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Step 2.1

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

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Step 2.1.1

Subtract $3$ from both sides of the inequality.

$a > 2$ or $- 4 a \geq 23 - 3$

Step 2.1.2

Subtract $3$ from $23$ .

$a > 2$ or $- 4 a \geq 20$

Step 2.2

Divide each term in $- 4 a \geq 20$ by $- 4$ and simplify.

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Step 2.2.1

Divide each term in $- 4 a \geq 20$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$a > 2$ or $\frac{- 4 a}{- 4} \leq \frac{20}{- 4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$a > 2$ or $\frac{- 4 a}{- 4} \leq \frac{20}{- 4}$

Step 2.2.2.1.2

Divide $a$ by $1$ .

$a > 2$ or $a \leq \frac{20}{- 4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $20$ by $- 4$ .

$a > 2$ or $a \leq - 5$

Step 3

The union consists of all of the elements that are contained in each interval.

$a \leq - 5$ or $a > 2$

Step 4

Convert the inequality to interval notation.

$(- \infty, - 5] \cup (2, \infty)$

Step 5