inequality involving modulus

OP

a picture of the question

Original post by mikeft

a picture of the question

First thing Id notice is what value must x be for the right hand side to be >=0 (why?).

Then the usual things to do are square up and remove the abs, though that will produce a quartic or /cubic or reason about the sign of the abs on each of the intervals defined by the factors. Think about the numerator factors (or not). A sketch may help for the numerator.

(edited 10 months ago)

Reply 3

OP

Original post by mqb2766

First thing Id notice is what value must x be for the right hand side to be >=0 (why?).

Then the usual things to do are square up and remove the modulus, though that will produce a quartic or /cubic or reason about the sign of the modulus on each of the intervals defined by the factors. Think about the numerator factors (or not). A sketch may help for the numerator.

were told not to square it there anything else?

Reply 4

Original post by mikeft

were told not to square it there anything else?

Did you investigate the numerator?

Personally, I agree about not squaring through. In some cases it can be simpler, but the usual way is to reason about the sign of the abs function on each of the intervals defined by the factors.

(edited 10 months ago)

Reply 5

OP

Original post by mqb2766

Did you investigate the numerator?

Personally, I agree about not squaring through. In some cases it can be simpler, but the usual way is to reason about the sign of the modulus on each of the intervals defined by the factors.

yes it has imaginary solutions

Reply 6

Original post by mikeft

yes it has imaginary solutions

Correct (complex) but what does it mean in the context of this question which is all about real values Id guess?

Reply 7

OP

Original post by mqb2766

Correct (complex) but what does it mean in the context of this question which is all about real values Id guess?

um i dont really. understand we have complex solutions what real values?

Reply 8

Original post by mikeft

um i dont really. understand we have complex solutions what real values?

Does the |.| represent the absolute value of real numbers or the modulus of complex numbers? The inequality only makes sense if its the former, so its got nothing to do with imaginary/complex roots.

What real factors does the numerator have, or not, and what can you say about it in relation to the absolute value function?

(edited 10 months ago)

Reply 9

OP

Original post by mqb2766

Does the |.| represent the absolute value of real numbers or the modulus of complex numbers? The inequality only makes sense if its the former, so its got nothing to do with imaginary/complex roots.

What real factors does the numerator have, or not, and what can you say about it in relation to the absolute value function?

i dont think it has real factors
what does the underlined mean?

Reply 10

Original post by mikeft

i dont think it has real factors
what does the underlined mean?

I agree the quadratic doesnt have real factors so it must be single signed, so is always positive or negative and therefore how does this knowledge simplify the absolute value function on the left hand side.

The bold answers the underlined part. Once you reason about the numerator quadratic appropriately, the inequality should be relatively simple.

In the context of the OP, sketching the function (sketching the numerator and denominator seperately) should have illustrated all this.

Reply 11

OP

Original post by mqb2766

I agree the quadratic doesnt have real factors so it must be single signed, so is always positive or negative and therefore how does this knowledge simplify the absolute value function on the left hand side.

The bold answers the underlined part. Once you reason about the numerator quadratic appropriately, the inequality should be relatively simple.

In the context of the OP, sketching the function (sketching the numerator and denominator seperately) should have illustrated all this.

i think its better to leave it so i can discuss it with my teacher tomorrow thank you though

Reply 12