8|y-4|>= 82 — step-by-step solution

1) Let's solve in

ℝ

8 \cdot | y - 4 | \geq 82

Since

2 > 0

, divide both sides by

2

without reversing the inequality:

4 \cdot | y - 4 | \geq 41

| A | \geq B

is equivalent to:

A \geq B

A \leq - B

2) Let's solve in

ℝ

y - 4 \geq \frac{41}{4}

ⓘ We add

4

to both sides

y \geq \frac{41}{4} + 4

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

\frac{41}{4} + 4

\frac{41 + 4 \cdot 4}{4}

y \geq \frac{41 + 4 \cdot 4}{4}

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 4

16

y \geq \frac{41 + 16}{4}

ⓘ The numerator and the denominator are computed before the division:

41 + 16

57

Solution found:

y \geq \frac{57}{4}

3) Let's solve in

ℝ

y - 4 \leq - \frac{41}{4}

ⓘ We add

4

to both sides

y \leq - \frac{41}{4} + 4

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

- \frac{41}{4} + 4

\frac{- 41 + 4 \cdot 4}{4}

y \leq \frac{- 41 + 4 \cdot 4}{4}

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 4

16

y \leq \frac{- 41 + 16}{4}

ⓘ The numerator and the denominator are computed before the division:

- 41 + 16

-25

Solution obtained:

y \leq - \frac{25}{4}

▶Solution:

y \in (-∞, - \frac{25}{4}] \cup [\frac{57}{4}, +∞)

4) The numeric value of:

y \in (-∞, - \frac{25}{4}] \cup [\frac{57}{4}, +∞)

equals:

▷

y \in (-∞, -6.25] \cup [14.25, +∞)