Solve the System of Inequalities 15<-2x+1 and -1<-2x+1

$15 < - 2 x + 1$ and $- 1 < - 2 x + 1$

Step 1

Simplify the first inequality.

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

$- 2 x + 1 > 15$ and $- 1 < - 2 x + 1$

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

Subtract $1$ from both sides of the inequality.

$- 2 x > 15 - 1$ and $- 1 < - 2 x + 1$

Subtract $1$ from $15$ .

$- 2 x > 14$ and $- 1 < - 2 x + 1$

Divide each term in $- 2 x > 14$ by $- 2$ and simplify.

Divide each term in $- 2 x > 14$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{14}{- 2}$ and $- 1 < - 2 x + 1$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{14}{- 2}$ and $- 1 < - 2 x + 1$

Divide $x$ by $1$ .

$x < \frac{14}{- 2}$ and $- 1 < - 2 x + 1$

Simplify the right side.

Divide $14$ by $- 2$ .

$x < - 7$ and $- 1 < - 2 x + 1$

Step 2

Simplify the second inequality.

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

$x < - 7$ and $- 2 x + 1 > - 1$

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

Subtract $1$ from both sides of the inequality.

$x < - 7$ and $- 2 x > - 1 - 1$

Subtract $1$ from $- 1$ .

$x < - 7$ and $- 2 x > - 2$

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

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

$x < - 7$ and $\frac{- 2 x}{- 2} < \frac{- 2}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$x < - 7$ and $\frac{- 2 x}{- 2} < \frac{- 2}{- 2}$

Divide $x$ by $1$ .

$x < - 7$ and $x < \frac{- 2}{- 2}$

Simplify the right side.

Divide $- 2$ by $- 2$ .

$x < - 7$ and $x < 1$

Step 3

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

$x < - 7$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x < - 7$

Interval Notation:

$(- \infty, - 7)$

Step 5

Solve the System of Inequalities 15<-2x+1 and -1<-2x+1

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