Solve the System of Inequalities 3-2x>=5 or 3(x-2)+1>7

$3 - 2 x \geq 5$ or $3 (x - 2) + 1 > 7$

Step 1

Simplify the first inequality.

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

Subtract $3$ from both sides of the inequality.

$- 2 x \geq 5 - 3$ or $3 (x - 2) + 1 > 7$

Subtract $3$ from $5$ .

$- 2 x \geq 2$ or $3 (x - 2) + 1 > 7$

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

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

$\frac{- 2 x}{- 2} \leq \frac{2}{- 2}$ or $3 (x - 2) + 1 > 7$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{2}{- 2}$ or $3 (x - 2) + 1 > 7$

Divide $x$ by $1$ .

$x \leq \frac{2}{- 2}$ or $3 (x - 2) + 1 > 7$

Simplify the right side.

Divide $2$ by $- 2$ .

$x \leq - 1$ or $3 (x - 2) + 1 > 7$

Step 2

Simplify the second inequality.

Simplify $3 (x - 2) + 1$ .

Simplify each term.

Apply the distributive property.

$x \leq - 1$ or $3 x + 3 \cdot - 2 + 1 > 7$

Multiply $3$ by $- 2$ .

$x \leq - 1$ or $3 x - 6 + 1 > 7$

Add $- 6$ and $1$ .

$x \leq - 1$ or $3 x - 5 > 7$

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

Add $5$ to both sides of the inequality.

$x \leq - 1$ or $3 x > 7 + 5$

Add $7$ and $5$ .

$x \leq - 1$ or $3 x > 12$

Divide each term in $3 x > 12$ by $3$ and simplify.

Divide each term in $3 x > 12$ by $3$ .

$x \leq - 1$ or $\frac{3 x}{3} > \frac{12}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$x \leq - 1$ or $\frac{3 x}{3} > \frac{12}{3}$

Divide $x$ by $1$ .

$x \leq - 1$ or $x > \frac{12}{3}$

Simplify the right side.

Divide $12$ by $3$ .

$x \leq - 1$ or $x > 4$

Step 3

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 1 or x > 4$

Interval Notation:

$(- \infty, - 1] \cup (4, \infty)$

Step 4

Solve the System of Inequalities 3-2x>=5 or 3(x-2)+1>7

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