If x is an integer, is 2x less than 80? HELP MATH TROUBLE

[benkriger]

I am having trouble with the math used in the explanation. Please read:

If x is an integer, is 2^x less than 80?

(1) 4^x-1 < 4^x - 120

(2) x^2 = 36
-----------
Since 26 = 64 and 27 = 128, 2x will be less than 80 only if the integer x is less than 7. So, we can rephrase the question as follows: "Is x < 7?"

The answer is (B), but, but when you read the explanation I have trouble following the math for statement one.


(1) INSUFFICIENT: We can solve the inequality for x.

4^x-1 < 4^x - 120
4^x-1 - 4^x < -120
4^x(4^-1) - 4^x < -120
4^x(1/4) - 4^x < -120
4^x[(1/4) - 1] < -120
4^x(-3/4) < -120
4^x > 160

How do you go from

4^x(1/4) - 4^x < -120

to

4^x[(1/4) - 1] < -120


Where does the 1 come from?


Since 43 = 64 and 44 = 256, x must be greater than 3. However, this is not enough to determine whether x < 7.

(2) SUFFICIENT: If x^2 = 36, then x = 6 or -6. Either way, this is enough to determine whether x < 7.

The correct answer is B.

[esledge]

How do you go from

4^x(1/4) - 4^x < -120

to

4^x[(1/4) - 1] < -120


Where does the 1 come from?

It comes from the fact that 4^x = 1 * 4^x. When you factor the 4^x out, you are dividing each term by 4^x and 4^x/4^x = 1.

Just to show another way of manipulating, the following was more intuitive to me. I don't like fractions as much as whole numbers, so I like to get rid of them ASAP:

4^(x-1) < 4^x - 120
(4^x)(4^-1) < 4^x - 120
(4^x)/4 < 4^x - 120
4^x < 4*4^x - 480
480 < 4*4^x - 4^x = 3*4^x
480/3 < 4^x
160 < 4^x

[mailtobook]

Gi there,

The one thing Idont get is why do we reverse de inequality in the last step:


4^x(-3/4) < -120
4^x > 160



I understand that we are to divide both sides by (-3/4), but wouldnt the value on the right be positive then? -120*(-4/3)

If so, then why do we need to reverse the symbol in this case?

thanks

[jnelson0612]

Gi there,

The one thing Idont get is why do we reverse de inequality in the last step:


4^x(-3/4) < -120
4^x > 160



I understand that we are to divide both sides by (-3/4), but wouldnt the value on the right be positive then? -120*(-4/3)

If so, then why do we need to reverse the symbol in this case?

thanks

Mailtobook,
Anytime you multiply or divide by a negative number across an inequality, you must reverse the inequality sign. Then you just carry on and calculate out your numbers. -120 *(-4/3) is 160.