How do you find the numbers such that four less than half the number is at least five and at most ten?

Oct 29, 2016

The number is any value between $18$ and $28$, inclusive.

Explanation:

"$4$ less than half the number" is written algebraically as :

$\frac{x}{2} - 4$.

Since this expression "is at least $5$", it is greater than or equal to $5$, and since it is "at most $10$", it is less than or equal to $10$. Putting all this together gives the compound inequality:

$5 \le \frac{x}{2} - 4 \le 10$

Remember that to simplify this type of inequality, your objective is to isolate the variable in the center section of the statement. We will accomplish this by adding $4$ to each section and then multiplying the entire inequality by $2$.

$5 + 4 \le \frac{x}{2} - 4 + 4 \le 10 + 4$

$9 \le \frac{x}{2} \le 14$

$2 \left(9 \le \frac{x}{2} \le 14\right)$

$18 \le x \le 28$

Therefore the number can be any number between $18$ and $28$, inclusive.

Oct 29, 2016

$18 \le x \le 28$ i.e $x$ is between $18$ and $28$

Explanation:

Let the number be $x$,

then four less than half the number is $\frac{x}{2} - 4$

and as it is at least $5$, we have $\frac{x}{2} - 4 \ge 5$

or $\frac{x}{2} \ge 5 + 4$ i.e. $\frac{x}{2} \ge 9$ and $x \ge 18$

Further as $\frac{x}{2} - 4$ is at most $10$, we have $\frac{x}{2} - 4 \le 10$

or $\frac{x}{2} \le 10 + 4$ i.e. $\frac{x}{2} \le 14$ and $x \le 28$

Combining the two we have $18 \le x \le 28$

i.e $x$ is between $18$ and $28$