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QuestionSolve the inequality ∣9n+2∣+9≥47|9 n + 2| + 9 \geq 47.

Studdy Solution

STEP 1

Assumptions1. The given inequality is ∣9n+∣+9≥47|9n+|+9 \geq47 . We need to solve for the variable nn

STEP 2

First, we simplify the inequality by subtracting9 from both sides.
∣9n+2∣+9−9≥47−9|9n+2|+9-9 \geq47-9

STEP 3

After simplifying, we get∣9n+2∣≥38|9n+2| \geq38

STEP 4

The absolute value inequality ∣a∣≥b|a| \geq b can be rewritten as a≥ba \geq b or a≤−ba \leq -b. So we can rewrite our inequality as two separate inequalities 9n+2≥389n+2 \geq38 and 9n+2≤−389n+2 \leq -38.

STEP 5

Let's solve the first inequality 9n+2≥389n+2 \geq38 first. We start by subtracting2 from both sides.
9n+2−2≥38−29n+2-2 \geq38-2

STEP 6

After simplifying, we get9n≥369n \geq36

STEP 7

Next, we divide both sides by9 to solve for nn.
n≥369n \geq \frac{36}{9}

STEP 8

After simplifying, we getn≥4n \geq4

STEP 9

Now, let's solve the second inequality 9n+2≤−389n+2 \leq -38. We start by subtracting2 from both sides.
9n+2−2≤−38−29n+2-2 \leq -38-2

STEP 10

After simplifying, we get9n≤−409n \leq -40

STEP 11

Next, we divide both sides by9 to solve for nn.
n≤−409n \leq \frac{-40}{9}

STEP 12

After simplifying, we getn≤−409n \leq -\frac{40}{9}

STEP 13

So, the solution to the inequality ∣9n+2∣+9≥47|9n+2|+9 \geq47 is n≥n \geq or n≤−409n \leq -\frac{40}{9}.

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