|(x-2)/(3x+5)|<= 4 — step-by-step solution

1) Let's solve in

ℝ

| \frac{x - 2}{3 x + 5} | \leq 4

| A | \leq B

is equivalent to

- B \leq A \leq B

2) Let's solve in

ℝ

\frac{x - 2}{3 x + 5} \leq 4

3) Let's sum the fractions:

\frac{x - 2}{3 x + 5} - 4 \leq 0

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{x - 2}{3 x + 5} - 4

\frac{(x - 2) - 4 \cdot (3 x + 5)}{3 x + 5}

\frac{(x - 2) - 4 \cdot (3 x + 5)}{3 x + 5} \leq 0

\frac{x - 2 - 4 \cdot (3 x + 5)}{3 x + 5} \leq 0

We study the sign of the numerator

Let's study the sign of

x - 2 - 4 \cdot (3 x + 5)

by solving in

ℝ

x - 2 - 4 \cdot (3 x + 5) \leq 0

ⓘ Expand by multiplying each term inside the parentheses:

- 4 \cdot (3 x + 5)

- (12 x + 20)

x - 2 - (12 x + 20) \leq 0

ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (12 x + 20)

- 12 x - 20

x

- 2

- 12 x

- 20 \leq 0

ⓘ Start from

-2

and move back

20

more on the number line, away from zero:

- 2 - 20

-22

■Like terms are grouped by adding their coefficients:

x - 12 x

-11 x

- 11 x - 22 \leq 0

ⓘ We add

22

to both sides

- 11 x \leq 22

We divide both sides by

-11

; since this number is negative, we reverse the inequality sign

x \geq \frac{22}{-11}

ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{22}{-11}

- \frac{22}{11}

x \geq - \frac{22}{11}

We divide the numerator and denominator of

\frac{22}{11}

by the common factor:

11

, which gives the new expression:

2

Candidates, pending the domain check:

x \geq -2

⇥So the expression

x - 2 - 4 \cdot (3 x + 5)

is negative or zero on

[-2, +∞)

and positive otherwise

We study the sign of the denominator, which must not be zero

⇥The expression

3 x + 5

is zero at:

⇥

x = - \frac{5}{3}

⇥Its coefficient of

x

3

, being positive, it is negative before and positive after

- \frac{5}{3}

(forbidden value)

4) Let's draw up the sign table:

x	$-∞$		$-2$		$- \frac{5}{3}$		$+∞$
$x - 2 - 4 \cdot (3 x + 5)$		+	0	-		-
$3 x + 5$		-		-	0	+
$\frac{x - 2}{3 x + 5} - 4$		-	0	+	❌	-

Candidates, pending the domain check:

x \in (-∞, -2] \cup (- \frac{5}{3}, +∞)

5) Let's solve in

ℝ

\frac{x - 2}{3 x + 5} \geq -4

6) Let's sum the fractions:

\frac{x - 2}{3 x + 5} + 4 \geq 0

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{x - 2}{3 x + 5} + 4

\frac{(x - 2) + 4 \cdot (3 x + 5)}{3 x + 5}

\frac{(x - 2) + 4 \cdot (3 x + 5)}{3 x + 5} \geq 0

\frac{x - 2 + 4 \cdot (3 x + 5)}{3 x + 5} \geq 0

We study the sign of the numerator

Let's study the sign of

x - 2 + 4 \cdot (3 x + 5)

by solving in

ℝ

x - 2 + 4 \cdot (3 x + 5) \geq 0

ⓘ Expand by multiplying each term inside the parentheses:

4 \cdot (3 x + 5)

12 x + 20

x

- 2

+ 12 x

+ 20 \geq 0

ⓘ Start from

-2

and move forward

20

on the number line, going past zero:

- 2 + 20

18

■Like terms are grouped by adding their coefficients:

x + 12 x

13 x

13 x + 18 \geq 0

13 x \geq -18

ⓘ Both sides are divided by

13

Candidates, pending the domain check:

x \geq - \frac{18}{13}

⇥So the expression

x - 2 + 4 \cdot (3 x + 5)

is positive or zero on

[- \frac{18}{13}, +∞)

and negative otherwise

We study the sign of the denominator, which must not be zero

⇥The expression

3 x + 5

is zero at:

⇥

x = - \frac{5}{3}

⇥Its coefficient of

x

3

, being positive, it is negative before and positive after

- \frac{5}{3}

(forbidden value)

7) Let's draw up the sign table:

x	$-∞$		$- \frac{5}{3}$		$- \frac{18}{13}$		$+∞$
$x - 2 + 4 \cdot (3 x + 5)$		-		-	0	+
$3 x + 5$		-	0	+		+
$\frac{x - 2}{3 x + 5} + 4$		+	❌	-	0	+

Candidates, pending the domain check:

x \in (-∞, - \frac{5}{3}) \cup [- \frac{18}{13}, +∞)

Let's build the table of the different terms to determine the global solution:

$x$	$-∞$		$-2$		$- \frac{5}{3}$		$- \frac{18}{13}$		$+∞$
$\frac{x - 2}{3 x + 5} \geq -4$		✅	✅	✅	❌	❌	✅	✅
$\frac{x - 2}{3 x + 5} \leq 4$		✅	✅	❌	❌	✅	✅	✅
Overall solution		✅	✅	❌	❌	❌	✅	✅

Candidates, pending the domain check:

x \in (-∞, -2] \cup [- \frac{18}{13}, +∞)

8) Let's find the domain of

| \frac{x - 2}{3 x + 5} | \leq 4

For the expression

\frac{x - 2}{3 x + 5}

to be defined, we must have:

3 x + 5 \neq 0

ⓘ We subtract

5

from both sides

3 x \neq -5

ⓘ Both sides are divided by

3

Domain constraints:

x \neq - \frac{5}{3}

▶Solution:

x \in (-∞, -2] \cup [- \frac{18}{13}, +∞)

9) The numeric value of:

x \in (-∞, -2] \cup [- \frac{18}{13}, +∞)

is:

▷

x \in (-∞, -2] \cup [\approx -1.3846, +∞)