Question

Which number is
n(n-3)<54

Answer 1

Explanation:

we can simplify the inequality by expanding the left side:

n(n-3) < 54

n^2 - 3n < 54

Next, we can move all the terms to one side of the inequality:

n^2 - 3n - 54 < 0

Now we need to find the values of n that make this quadratic expression less than zero. We can use factoring or the quadratic formula to find the roots of the expression, which are:

n = -6 and n = 9

These roots divide the number line into three intervals: (-∞,-6), (-6,9), and (9,∞). We can test a value in each interval to determine if the expression is less than zero in that interval:

If we plug in n = -7, we get (-7)(-10) = 70, which is greater than zero. So the expression is not less than zero in the interval (-∞,-6).

If we plug in n = 0, we get (0)(-3) = 0, which is not less than zero. So the expression is not less than zero in the interval (-6,9).

If we plug in n = 10, we get (10)(7) = 70, which is greater than zero. So the expression is not less than zero in the interval (9,∞).

Therefore, the solution to the inequality is the interval (-6,9). In interval notation, we can write:

-6 < n < 9

So, any value of n that falls between -6 and 9 (excluding -6 and 9) will make the inequality n(n-3)<54 true.

Answer 2

To solve the inequality n(n-3) < 54, we can first rearrange it into standard quadratic form by bringing all the terms to one side:

n^2 - 3n - 54 < 0

Next, we can factor the quadratic expression:

(n - 9)(n + 6) < 0

To determine the values of n that satisfy this inequality, we can use the sign test. We evaluate the expression (n - 9)(n + 6) for some test values of n in each of the intervals (-∞, -6), (-6, 9), and (9, ∞) to determine the sign of the expression in each interval.

For example, if we choose n = -7 (which is in the interval (-∞, -6)), then we have:

(-7 - 9)(-7 + 6) = (-16)(-1) = 16 > 0

This tells us that the expression (n - 9)(n + 6) is positive in the interval (-∞, -6).

If we choose n = 0 (which is in the interval (-6, 9)), then we have:

(0 - 9)(0 + 6) = (-9)(6) = -54 < 0

This tells us that the expression (n - 9)(n + 6) is negative in the interval (-6, 9).

If we choose n = 10 (which is in the interval (9, ∞)), then we have:

(10 - 9)(10 + 6) = (1)(16) = 16 > 0

This tells us that the expression (n - 9)(n + 6) is positive in the interval (9, ∞).

Since the inequality (n - 9)(n + 6) < 0 is true only in the interval (-6, 9), we can conclude that the solution to the inequality n(n-3) < 54 is:

-6 < n < 9

Therefore, the number n that satisfies the inequality is any number between -6 and 9 (exclusive).