Algebra Examples

$3 x - 7 \leq 5 x + 5$ and $- 4 x + 8 < 5 x - 1$

Step 1

Simplify the first inequality.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Subtract $5 x$ from both sides of the inequality.

$3 x - 7 - 5 x \leq 5$ and $- 4 x + 8 < 5 x - 1$

Step 1.1.2

Subtract $5 x$ from $3 x$ .

$- 2 x - 7 \leq 5$ and $- 4 x + 8 < 5 x - 1$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Add $7$ to both sides of the inequality.

$- 2 x \leq 5 + 7$ and $- 4 x + 8 < 5 x - 1$

Step 1.2.2

Add $5$ and $7$ .

$- 2 x \leq 12$ and $- 4 x + 8 < 5 x - 1$

Step 1.3

Divide each term in $- 2 x \leq 12$ by $- 2$ and simplify.

Tap for more steps...

Step 1.3.1

Divide each term in $- 2 x \leq 12$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{12}{- 2}$ and $- 4 x + 8 < 5 x - 1$

Step 1.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{12}{- 2}$ and $- 4 x + 8 < 5 x - 1$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{12}{- 2}$ and $- 4 x + 8 < 5 x - 1$

Step 1.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.1

Divide $12$ by $- 2$ .

$x \geq - 6$ and $- 4 x + 8 < 5 x - 1$

Step 2

Simplify the second inequality.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

Subtract $5 x$ from both sides of the inequality.

$x \geq - 6$ and $- 4 x + 8 - 5 x < - 1$

Step 2.1.2

Subtract $5 x$ from $- 4 x$ .

$x \geq - 6$ and $- 9 x + 8 < - 1$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Subtract $8$ from both sides of the inequality.

$x \geq - 6$ and $- 9 x < - 1 - 8$

Step 2.2.2

Subtract $8$ from $- 1$ .

$x \geq - 6$ and $- 9 x < - 9$

Step 2.3

Divide each term in $- 9 x < - 9$ by $- 9$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $- 9 x < - 9$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$x \geq - 6$ and $\frac{- 9 x}{- 9} > \frac{- 9}{- 9}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $- 9$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$x \geq - 6$ and $\frac{- 9 x}{- 9} > \frac{- 9}{- 9}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x \geq - 6$ and $x > \frac{- 9}{- 9}$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Divide $- 9$ by $- 9$ .

$x \geq - 6$ and $x > 1$

Step 3

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

$x > 1$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x > 1$

Interval Notation:

$(1, \infty)$

Step 5