Question

Question:-
If x belongs to I, A is the solution set of 2(x - 1) < 3x - 1 and B is the solution set of 4x - 3<=8 + x, find A intersection B.
_
•Urgent Help Request!!
•No Spamming/irrelevant comments please​

Answer 1

Kindly refer to given attachment

#helpingismypleasure

Answer 2

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept}}}$

Here the concept of Linear Inequalities have been used. We see we are given the two inequalities and we need to find A B. Firstly we shall solve the inequality of both cases separately. Then we will get two values. That two values are terminating values of the set. Now we will find the solution which will represent A ∩ B.

Let's do it !!

________________________________________________

★ Solution :-

Given,

» x belongs to I

This means, x ϵ Integers

This means that value of x should be integers only not fractional part.

» A is solution set of 2(x - 1) < 3x - 1

» B is solution set of 4x - 3 ≤ 8 + x

________________________________________________

~ For the solution of First Inequality ::

Clearly, we see for the first inequality A is the solution set. Even x is the solution of this inequality. So whatever value of x we will get will be the solution of set of this inequality and will belong to A.

We are given that,

$\\\;\bf{\rightarrow\;\;\green{2(x\:-\:1)\;<\;3x\;-\;1}}$

Now simplifying this, we get

$\\\;\sf{\rightarrow\;\;2x\:-\:2\;<\;3x\;-\;1}$

$\\\;\sf{\rightarrow\;\;2x\:-\:2\;<\;3x\;-\;1}$

Subtracting 2x from both sides, we get

$\\\;\sf{\rightarrow\;\;2x\:-\:2\;-\;2x\;<\;3x\;-\;1\;-\;2x}$

$\\\;\sf{\rightarrow\;\;-\:2\;<\;x\;-\;1}$

Adding 1 on both sides, we get

$\\\;\sf{\rightarrow\;\;-\:2\;+\;1\;<\;x\;-\;1\;+\;1}$

$\\\;\bf{\rightarrow\;\;\red{-\:1\;\;<\;\;x}}$

Here we got one value of x. This is even a terminal point in set value of A ∩ B.

Let this be equation a.)

________________________________________________

~ For the solution of Second Inequality ::

Similarly like above, B is solution set of this inequality. So the value of x will also belong to the set B.

We are given that,

$\\\;\bf{\rightarrow\;\;\orange{4x\;-\;3\;\leq\;8\;+\;x}}$

On simplifying this, we get

$\\\;\sf{\rightarrow\;\;4x\;-\;3\;\leq\;8\;+\;x}$

Now subtracting x from both sides, we get

$\\\;\sf{\rightarrow\;\;4x\;-\;3\;-\;x\;\leq\;8\;+\;x\;-\;x}$

$\\\;\sf{\rightarrow\;\;3x\;-\;3\;\leq\;8}$

Now adding 3 to both sides, we get

$\\\;\sf{\rightarrow\;\;3x\;-\;3\;+\;3\;\leq\;8\;+\;3}$

$\\\;\sf{\rightarrow\;\;3x\;\;\leq\;\;11}$

Dividing both sides by 3, we get

$\\\;\sf{\rightarrow\;\;\dfrac{3x}{3}\;\;\leq\;\;\dfrac{11}{3}}$

$\\\;\sf{\rightarrow\;\;x\;\;\leq\;\;\dfrac{9\;+\;2}{3}}$

Separating numerators, we get

$\\\;\sf{\rightarrow\;\;x\;\;\leq\;\;\dfrac{9}{3}\;+\;\dfrac{2}{3}}$

$\\\;\bf{\rightarrow\;\;x\;\;\leq\;\;3\;+\;\dfrac{2}{3}}$

We see that, here x ≤ 3 + ⅔ . From above we know that x ϵ I so we can neglect the fractional part. Even ⅔ < 1 . So, we can neglect this value and write the above equation as,

$\\\;\bf{\rightarrow\;\;\blue{x\;\;\leq\;\;3}}$

Here we got another solution set of x. This is also a terminal point in the set value of A ∩ B.

Let this be equation b.)

________________________________________________

~ For the set of A ∩ B ::

We see that A is solution set of first inequality and B is solution set of second inequality. x denotes the solution of both inequality.

We need to find the value of A ∩ B.

So let's combine both the equations we got.

$\\\;\bf{\rightarrow\;\;\gray{-\:1\;\;<\;\;x\;\;\leq\;\;3}}$

We see that the value of x is greater than -1 but less than or equal to 3 . This will give us, A ∩ B shows the common value between teh solution set of A and B that is solution of both inequalities.

$\\\;\bf{\rightarrow\;\;\pink{x\;\;\epsilon\;\;(-\:1,\;3]}}$

This shows that -1 is not included in this set.

So possible values of x can be 0, 1, 2 and 3 .

Hence, we get,

$\\\;\displaystyle{\bf{\purple{\mapsto\;\;A\;\cap\;B\;=\;\{0,\;1,\;2,\;3\}}}}$

$\\\;\boxed{\underline{\tt{Hence,\;\;required\;\;solution\;\;set\;=\;\bf{\purple{\{0,\;1,\;2,\;3\}}}}}}$

________________________________________________

★ More to know :-

• Linear Inequalities : These are the expressions which don't have any solution which is the exact value of variable. These show the range of the value of variable.