Algebra Examples

\frac{a}{- 2} < - 1

$\frac{a}{- 2} < - 1$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1

Simplify the first inequality.

Tap for more steps...

Step 1.1

Multiply both sides by

- 2

$- 2$ .

\frac{a}{- 2} \cdot - 2 > 2

$\frac{a}{- 2} \cdot - 2 > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Simplify the left side.

Tap for more steps...

Step 1.2.1.1

Simplify

\frac{a}{- 2} \cdot - 2

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

Tap for more steps...

Step 1.2.1.1.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 1.2.1.1.1.1

Factor

2

$2$ out of

- 2

$- 2$ .

\frac{a}{2 (- 1)} \cdot - 2 > 2

$\frac{a}{2 (- 1)} \cdot - 2 > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.1.2

Factor

2

$2$ out of

- 2

$- 2$ .

\frac{a}{2 \cdot - 1} \cdot (2 \cdot - 1) > 2

$\frac{a}{2 \cdot - 1} \cdot (2 \cdot - 1) > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.1.3

Cancel the common factor.

\frac{a}{2 \cdot - 1} \cdot (2 \cdot - 1) > 2

$\frac{a}{2 \cdot - 1} \cdot (2 \cdot - 1) > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.1.4

Rewrite the expression.

\frac{a}{- 1} \cdot - 1 > 2

$\frac{a}{- 1} \cdot - 1 > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

\frac{a}{- 1} \cdot - 1 > 2

$\frac{a}{- 1} \cdot - 1 > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.2

Combine

\frac{a}{- 1}

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

- 1

$- 1$ .

\frac{a \cdot - 1}{- 1} > 2

$\frac{a \cdot - 1}{- 1} > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.3

Simplify the expression.

Tap for more steps...

Step 1.2.1.1.3.1

Move the negative one from the denominator of

\frac{a \cdot - 1}{- 1}

$\frac{a \cdot - 1}{- 1}$ .

- 1 \cdot (a \cdot - 1) > 2

$- 1 \cdot (a \cdot - 1) > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.3.2

Rewrite

- 1 \cdot (a \cdot - 1)

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

- (a \cdot - 1)

$- (a \cdot - 1)$ .

- (a \cdot - 1) > 2

$- (a \cdot - 1) > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.3.3

Move

- 1

$- 1$ to the left of

a

$a$ .

- (- 1 \cdot a) > 2

$- (- 1 \cdot a) > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.3.4

Rewrite

- 1 a

$- 1 a$ as

- a

$- a$ .

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.4

Multiply

- - a

$- - a$ .

Tap for more steps...

Step 1.2.1.1.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

1 a > 2

$1 a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.1.1.4.2

Multiply

a

$a$ by

1

$1$ .

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 1.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.2.1

Multiply

- 1

$- 1$ by

- 2

$- 2$ .

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

a > 2

$a > 2$ or

- 4 a + 3 \geq 23

$- 4 a + 3 \geq 23$

Step 2

Simplify the second inequality.

Tap for more steps...

Step 2.1

Move all terms not containing

a

$a$ to the right side of the inequality.

Tap for more steps...

Step 2.1.1

Subtract

3

$3$ from both sides of the inequality.

a > 2

$a > 2$ or

- 4 a \geq 23 - 3

$- 4 a \geq 23 - 3$

Step 2.1.2

Subtract

3

$3$ from

23

$23$ .

a > 2

$a > 2$ or

- 4 a \geq 20

$- 4 a \geq 20$

a > 2

$a > 2$ or

- 4 a \geq 20

$- 4 a \geq 20$

Step 2.2

Divide each term in

- 4 a \geq 20

$- 4 a \geq 20$ by

- 4

$- 4$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in

- 4 a \geq 20

$- 4 a \geq 20$ by

- 4

$- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

a > 2

$a > 2$ or

\frac{- 4 a}{- 4} \leq \frac{20}{- 4}

$\frac{- 4 a}{- 4} \leq \frac{20}{- 4}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of

- 4

$- 4$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

a > 2

$a > 2$ or

\frac{- 4 a}{- 4} \leq \frac{20}{- 4}

$\frac{- 4 a}{- 4} \leq \frac{20}{- 4}$

Step 2.2.2.1.2

Divide

a

$a$ by

1

$1$ .

a > 2

$a > 2$ or

a \leq \frac{20}{- 4}

$a \leq \frac{20}{- 4}$

a > 2

$a > 2$ or

a \leq \frac{20}{- 4}

$a \leq \frac{20}{- 4}$

a > 2

$a > 2$ or

a \leq \frac{20}{- 4}

$a \leq \frac{20}{- 4}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide

20

$20$ by

- 4

$- 4$ .

a > 2

$a > 2$ or

a \leq - 5

$a \leq - 5$

a > 2

$a > 2$ or

a \leq - 5

$a \leq - 5$

a > 2

$a > 2$ or

a \leq - 5

$a \leq - 5$

a > 2

$a > 2$ or

a \leq - 5

$a \leq - 5$

Step 3

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

a \leq - 5

$a \leq - 5$ or

a > 2

$a > 2$

Step 4

Convert the inequality to interval notation.

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

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

Step 5