-4<= (3x+11)/7 <3 — step-by-step solution

1) We have the following 2 conditions:

[1]:

-4 \leq \frac{3 x + 11}{7}

[2]:

\frac{3 x + 11}{7} < 3

2) Let's solve for x:

-4 \leq \frac{3 x + 11}{7}

[1]

\frac{3 x + 11}{7} \geq -4

3 x + 11 \geq -4 \cdot 7

ⓘ A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

-4 \cdot 7

-28

3 x + 11 \geq -28

ⓘ We subtract

11

from both sides

3) Let's calculate:

3 x \geq - 28 - 11

ⓘ Start from

-28

and move back

11

more on the number line, away from zero:

- 28 - 11

-39

3 x \geq -39

ⓘ Both sides are divided by

3

x \geq - \frac{39}{3}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

3

, their greatest common divisor:

- \frac{39}{3}

-13

Solution obtained [1]:

x \geq -13

4) Let's solve for x:

\frac{3 x + 11}{7} < 3

[2] using

x \geq -13

\frac{3 x + 11}{7} < 3

3 x + 11 < 3 \cdot 7

ⅰ

3 \cdot 7

21

3 x + 11 < 21

ⓘ We subtract

11

from both sides

3 x < 21 - 11

ⓘ Start from

21

and move back

11

on the number line:

21 - 11

10

3 x < 10

ⓘ Both sides are divided by

3

Solution found [2]:

x < \frac{10}{3}

Intersecting

x \geq -13

and

x < \frac{10}{3}

gives

x \in [-13, \frac{10}{3})

▶Solution:

x \in [-13, \frac{10}{3})

5) The numeric value of:

x \in [-13, \frac{10}{3})

is:

▷

x \in [-13, \approx 3.3333)