Solve the System of Inequalities 6x+1<8x+7<6x+9

$6 x + 1 < 8 x + 7 < 6 x + 9$

Step 1

Solve $6 x + 1 < 8 x + 7$ for $x$ .

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

Subtract $8 x$ from both sides of the inequality.

$6 x + 1 - 8 x < 7$

Subtract $8 x$ from $6 x$ .

$- 2 x + 1 < 7$

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

Subtract $1$ from both sides of the inequality.

$- 2 x < 7 - 1$

Subtract $1$ from $7$ .

$- 2 x < 6$

Divide each term in $- 2 x < 6$ by $- 2$ and simplify.

Divide each term in $- 2 x < 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{6}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{6}{- 2}$

Divide $x$ by $1$ .

$x > \frac{6}{- 2}$

Simplify the right side.

Divide $6$ by $- 2$ .

$x > - 3$

Step 2

Solve $8 x + 7 < 6 x + 9$ for $x$ .

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

Subtract $6 x$ from both sides of the inequality.

$8 x + 7 - 6 x < 9$

Subtract $6 x$ from $8 x$ .

$2 x + 7 < 9$

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

Subtract $7$ from both sides of the inequality.

$2 x < 9 - 7$

Subtract $7$ from $9$ .

$2 x < 2$

Divide each term in $2 x < 2$ by $2$ and simplify.

Divide each term in $2 x < 2$ by $2$ .

$\frac{2 x}{2} < \frac{2}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{2}{2}$

Divide $x$ by $1$ .

$x < \frac{2}{2}$

Simplify the right side.

Divide $2$ by $2$ .

$x < 1$

Step 3

Find the intersection of $x > - 3$ and $x < 1$ .

$- 3 < x < 1$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 3 < x < 1$

Interval Notation:

$(- 3, 1)$

Step 5

Solve the System of Inequalities 6x+1<8x+7<6x+9

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