Solve the System of Inequalities 34<3x+7 and 3x+7<=43

$34 < 3 x + 7$ and $3 x + 7 \leq 43$

Step 1

Simplify the first inequality.

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

$3 x + 7 > 34$ 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.

$3 x > 34 - 7$ and $3 x + 7 \leq 43$

Subtract $7$ from $34$ .

$3 x > 27$ and $3 x + 7 \leq 43$

Divide each term in $3 x > 27$ by $3$ and simplify.

Divide each term in $3 x > 27$ by $3$ .

$\frac{3 x}{3} > \frac{27}{3}$ and $3 x + 7 \leq 43$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} > \frac{27}{3}$ and $3 x + 7 \leq 43$

Divide $x$ by $1$ .

$x > \frac{27}{3}$ and $3 x + 7 \leq 43$

Simplify the right side.

Divide $27$ by $3$ .

$x > 9$ and $3 x + 7 \leq 43$

Step 2

Simplify the second inequality.

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

Subtract $7$ from both sides of the inequality.

$x > 9$ and $3 x \leq 43 - 7$

Subtract $7$ from $43$ .

$x > 9$ 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 $3$ .

$x > 9$ and $\frac{3 x}{3} \leq \frac{36}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$x > 9$ and $\frac{3 x}{3} \leq \frac{36}{3}$

Divide $x$ by $1$ .

$x > 9$ and $x \leq \frac{36}{3}$

Simplify the right side.

Divide $36$ by $3$ .

$x > 9$ and $x \leq 12$

Step 3

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

$9 < x \leq 12$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$9 < x \leq 12$

Interval Notation:

$(9, 12]$

Step 5

Solve the System of Inequalities 34<3x+7 and 3x+7<=43

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