Question

А) 3 11/15 < x/15 < 4 б) 3 1/8 < 25/x < 8 1/3

Answer 1

А) х>33
х<60
х€(33, 60)

б) х€(75/8, 200/3)
ОДЗ: х=\ 0
х€(75/8, 200/3)

Answer 2

To solve the given inequalities, we need to find the range of possible values for the variable x in each case. Let's solve them step by step:

А) 3 11/15 < x/15 < 4

To begin, we can convert the mixed numbers to improper fractions: 3 11/15 = (3 * 15 + 11) / 15 = 56/15 4 = 4/1

Now, the inequality becomes: 56/15 < x/15 < 4/1

To eliminate the denominators, we can multiply all the terms by 15 (the common denominator): 15 * (56/15) < 15 * (x/15) < 15 * (4/1)

Simplifying: 56 < x < 60

So the solution for (A) is x lies in the open interval (56, 60).

Б) 3 1/8 < 25/x < 8 1/3

First, convert the mixed numbers to improper fractions: 3 1/8 = (3 * 8 + 1) / 8 = 25/8 8 1/3 = (8 * 3 + 1) / 3 = 25/3

Now, the inequality becomes: 25/8 < 25/x < 25/3

To eliminate the denominators, we can multiply all the terms by 8 * 3 = 24 (the common denominator): 24 * (25/8) < 24 * (25/x) < 24 * (25/3)

Simplifying: 75 < 600/x < 200

Now, we can solve the individual inequalities:

1. 75 < 600/x

To isolate x, divide both sides by 75: 75 / 75 < (600/x) / 75 1 < 8/x

Now, we can invert the inequality (since we are dividing by a negative number, the inequality flips): x < 8

2. 600/x < 200

To isolate x, divide both sides by 200: (600/x) / 200 < 200 / 200 3 < 600/x

Now, invert the inequality: x < 600/3 x < 200

So the solution for (B) is x lies in the open interval (8, 200).

To summarize: (A) The solution for x is in the open interval (56, 60). (B) The solution for x is in the open interval (8, 200).