MYTUTOR SUBJECT ANSWERS

634 views

The inequality x^4 < 8x^2 + 9 is satisfied precisely when...

This is a multiple choice question, with possible answers:

(a) -3 < x <  3;

(b)  0 < x <  4;

(c)  1 < x <  3;

(d) -1 < x <  9;

(e) -3 < x < -1.

Let's start by rearranging the inequality to get

x^4 - 8x^2 - 9 < 0.

Now, we notice that x^4 = (x^2)^2, and so what we have on the left-hand side of our inequality above is really a quadratic equation in x^2:

(x^2)^2 - 8x^2 - 9 < 0.

So we can factor this like a normal quadratic: look for two numbers that add to make -8 and multiply to make -9. It turns out that -9 and +1 work, then our inequality is simply

(x^2 - 9)(x^2 + 1) < 0.

If we multiply two numbers together, the only way for the product to be negative (less than zero) is for one of the numbers to be negative and the other positive. But x^2 is positive no matter what value x takes, and so (x^2 + 1) is definitely going to be positive, for all values of x.

So we have to have (x^2 - 9) negative:

x^2 - 9 < 0.

And so

x^2 < 9,

hence

-3 < x < 3,

i.e. the answer is (a).

Tim H. A Level Maths tutor, Uni Admissions Test .MAT. tutor

1 year ago

Answered by Tim, an Uni Admissions Test .MAT. tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

6 SUBJECT SPECIALISTS

£30 /hr

Louis S.

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered: .MAT., Physics+ 6 more

.MAT.
Physics
Maths
Further Mathematics
Extended Project Qualification
.STEP.
-Personal Statements-
-Oxbridge Preparation-

“ second year undergraduate at the University of Cambridge, studying for a B.A. in Mathematics, having received A*s in A-Level Mathematics, Further Mathematics (Edexcel) and Physics (AQA)”

£25 /hr

Louise S.

Degree: Mathematics with Education & languages (Bachelors) - Imperial College London University

Subjects offered: .MAT., Spanish+ 4 more

.MAT.
Spanish
Physics
Maths
Further Mathematics
Chemistry

“I am a 2nd year maths student at imperial. My aim is for students to understand what they are learning, not just to learn it by rote. I think learning should be fun , and I will try and make them so.”

£26 /hr

Joe B.

Degree: Mathematics G100 (Bachelors) - Bath University

Subjects offered: .MAT., Maths+ 4 more

.MAT.
Maths
Further Mathematics
Economics
.STEP.
-Personal Statements-

“About Me Hi, I'm Joe, a first year mathematics student from Luton studying at Bath University. I am an accomplished mathematician and economist, having achieved A* grades in A Level Maths, Further Maths and Economics in June 2016. As ...”

About the author

Tim H.

Currently unavailable: for new students

Degree: Mathematics (Masters) - Oxford, Hertford College University

Subjects offered: .MAT., Maths

.MAT.
Maths

“Fourth-year undergraduate at Oxford, studying for a Masters in Maths. Eager to help people understand maths, not just memorise it.”

You may also like...

Other Uni Admissions Test .MAT. questions

The sequence xn is given by the formula x_n = n^3 − 9n^2 + 631. What is the largest value of n for which x_n > x_(n+1)?

When is the inequality x^4 < 8x^2 + 9 true?

The inequality x^4 < 8x^2 + 9 is satisfied precisely when...

How many distinct real roots does the equation x^3 − 30x^2 + 108x − 104 = 0 have?

View Uni Admissions Test .MAT. tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok