Solve the System of Inequalities 37<3x+7 and 43>=3x+7

$37 < 3 x + 7$ and $43 \geq 3 x + 7$

Simplify the first inequality.

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

$3 x + 7 > 37$ and $43 \geq 3 x + 7$

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

Subtract $7$ from both sides of the inequality.

$3 x > 37 - 7$ and $43 \geq 3 x + 7$

Subtract $7$ from $37$ .

$3 x > 30$ and $43 \geq 3 x + 7$

Divide each term in $3 x > 30$ by $3$ and simplify.

Divide each term in $3 x > 30$ by $3$ .

$\frac{3 x}{3} > \frac{30}{3}$ and $43 \geq 3 x + 7$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} > \frac{30}{3}$ and $43 \geq 3 x + 7$

Divide $x$ by $1$ .

$x > \frac{30}{3}$ and $43 \geq 3 x + 7$

Simplify the right side.

Divide $30$ by $3$ .

$x > 10$ and $43 \geq 3 x + 7$

Simplify the second inequality.

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

$x > 10$ and $3 x + 7 \leq 43$

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

Subtract $7$ from both sides of the inequality.

$x > 10$ and $3 x \leq 43 - 7$

Subtract $7$ from $43$ .

$x > 10$ and $3 x \leq 36$

Divide each term in $3 x \leq 36$ by $3$ and simplify.

Divide each term in $3 x \leq 36$ by $3$ .

$x > 10$ and $\frac{3 x}{3} \leq \frac{36}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$x > 10$ and $\frac{3 x}{3} \leq \frac{36}{3}$

Divide $x$ by $1$ .

$x > 10$ and $x \leq \frac{36}{3}$

Simplify the right side.

Divide $36$ by $3$ .

$x > 10$ and $x \leq 12$

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

$10 < x \leq 12$

The result can be shown in multiple forms.

Inequality Form:

$10 < x \leq 12$

Interval Notation:

$(10, 12]$

Solve the System of Inequalities 37<3x+7 and 43>=3x+7

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