Question

How do you solve the inequality \[10 - x \geqslant 6\] ?

Answer 1

Hint: In this question, we are given an inequality and we have been asked to find the value of \[x\] . You can either shift the terms to the other side such that only \[x\] remains on one side, or you can add or subtract the terms on both the sides, so that you get the required range. You won’t get any particular value of \[x\] , but you will get only a range.

Formula used: Addition, Subtraction, Multiplication and Division property of inequality.
If we add, subtract, multiply or divide any constant to the both of the sides of an inequality then the sign of the inequality does not change.

Complete step-by-step solution:
The given inequality in the above question is as following:
\[10 - x \geqslant 6\] .
Now, we will add \[x\] on the both sides of the inequality:
$\Rightarrow$\[10 - x + x \geqslant 6 + x\] . (According to addition property, the sign remain same)
$\Rightarrow$\[ \Rightarrow 10 \geqslant 6 + x.\]
Now, we will subtract \[6\] from both the sides, we get:
$\Rightarrow$\[10 - 6 \geqslant 6 + x - 6.\] (According to subtraction rule, the sign will remain same)
By solving the above inequality, we get:
$\Rightarrow$\[4 \geqslant x.\]

\[\therefore \]The values of \[x\] should be less than or equal to \[4\].

Note: Points to remember:
If we want to cross check the process that whether the answer is correct or not, we can put the values of \[x\] inside the domain as well as outside the domain.
Let's say, we consider the value of \[x\] is \[5\] .
Now, if we put this value into the inequality, we get:
\[10 - x \geqslant 6\]
\[ \Rightarrow 10 - 5 \geqslant 6\]
\[ \Rightarrow 5 \geqslant 6\]
But it contradicts the general rule of mathematics, as \[5\] is not greater than \[6\].
So, we can confirm that the value of \[x\] shall lie below \[4\] or equal to \[4\] .