Solve the System of Inequalities -2x+1>=-5 and 2x+1>3

$- 2 x + 1 \geq - 5$ and $2 x + 1 > 3$

Step 1

Simplify the first inequality.

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

Subtract $1$ from both sides of the inequality.

$- 2 x \geq - 5 - 1$ and $2 x + 1 > 3$

Subtract $1$ from $- 5$ .

$- 2 x \geq - 6$ and $2 x + 1 > 3$

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

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

$\frac{- 2 x}{- 2} \leq \frac{- 6}{- 2}$ and $2 x + 1 > 3$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 6}{- 2}$ and $2 x + 1 > 3$

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 2}$ and $2 x + 1 > 3$

Simplify the right side.

Divide $- 6$ by $- 2$ .

$x \leq 3$ and $2 x + 1 > 3$

Step 2

Simplify the second inequality.

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

Subtract $1$ from both sides of the inequality.

$x \leq 3$ and $2 x > 3 - 1$

Subtract $1$ from $3$ .

$x \leq 3$ and $2 x > 2$

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

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

$x \leq 3$ and $\frac{2 x}{2} > \frac{2}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$x \leq 3$ and $\frac{2 x}{2} > \frac{2}{2}$

Divide $x$ by $1$ .

$x \leq 3$ and $x > \frac{2}{2}$

Simplify the right side.

Divide $2$ by $2$ .

$x \leq 3$ and $x > 1$

Step 3

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

$1 < x \leq 3$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$1 < x \leq 3$

Interval Notation:

$(1, 3]$

Step 5

Solve the System of Inequalities -2x+1>=-5 and 2x+1>3

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