Question

Half the sum of a number, $x$ and $15$ is at most the sum of the opposite of twice the number and $1.25$. What is the range of possible values for the number?

Answer 1

Hint: In the above question, we are given an unknown number $x$. The sum of that number $x$ and $15$ is then reduced by half. On the other hand, the opposite of that number $x$, that is $–x$, is increased by twice, so that it becomes $-2x$. Now the number $-2x$ is added to another number $1.25$. That is another sum of numbers. Now it is given in the question that the first half sum in the question is at most the sum of other numbers. That means it is either equal or less than the other sum but it is not more than the sum of other numbers. Putting all this information in an equation, we can solve the equation for $x$. In that way, the range for possible values of $x$ can be obtained.

Complete step by step answer:
Given that, a number x.
It is added to 15, so the sum is,
\[ \Rightarrow x + 15\]
Now, the half of this sum is
\[ \Rightarrow \dfrac{{x + 15}}{2}\]
On the other hand, opposite of the number $x$, that is
\[ \Rightarrow - x\]

Twice the opposite of $x$, i.e.
\[ \Rightarrow - 2x\]
Sum of twice the opposite of x and 1.25, which is
\[ \Rightarrow - 2x + 1.25\]
According to the question, it is given that the first sum is at most the other sum.That means it is not more than the other sum, it is either less or equal to the other sum.Putting that information in an equation, that gives us the following equation
\[ \Rightarrow \dfrac{{x + 15}}{2} \leqslant - 2x + 1.25\]

Now we have to solve the equation for x.
So, multiplying both sides by 2, we get
\[ \Rightarrow x + 15 \leqslant - 4x + 2.5\]
Shifting the variables to LHS and the numbers to RHS, we get
\[ \Rightarrow x + 4x \leqslant 2.5 - 15\]
That gives,
\[ \Rightarrow 5x \leqslant - 12.5\]
Solving for x we get,
\[ \Rightarrow x \leqslant - \dfrac{{12.5}}{5}\]
Therefore,
\[ \therefore x \leqslant - 2.5\]
That is the required range of x.
So, $x$ is always smaller than or equal to -2.5.

Therefore, the range for possible values of x is \[R = \left\{ {x \in IR:x \leqslant - 2.5} \right\}\].

Note: In mathematical statements, if it is given that statement A is at most statement B, it means that statement A can't be more than statement B, and thus statement A is always lesser than or equal to statement B. Therefore, we can write the two statements together in a form of mathematical equation as,
\[ \Rightarrow A \leqslant B\]
In this way, we can use the information given in the statements in mathematical form.Therefore, after putting all the known information in the form of an equation, we can solve the equation for the required unknown value.