Question

Solve each inequality.

$\left|5-m\right|<1$

Short answer

The required solution of the given inequality is $4<m<6$ or $\left(4,6\right)$.

Step-by-step solution

– Properties of inequality.

In an absolute inequality, if $\left|x\right|<a$, then $-a<x<a$.

According to the subtraction property of inequality, if $a<b$, then $a-c<b-c$, where a,b,c are real values.

According to the multiplication property of inequality, if the inequality a<b multiplied by -1, then -a>-b, where a,b,c are real values.

– Given information.

The given inequality is:

$\left|5-m\right|<1$

– Use the properties of inequality to solve the given inequality.

$\left|5-m\right|<1$

It is an absolute inequality and it can be written as:

$-1<5-m<1$

Subtract 5 from each side of the above inequality.

$\begin{array}{c}-1-5<5-m-5<1-5\\ -6<-m<-4\end{array}$

Multiply each side by -1.

$\begin{array}{c}-6(-1)>-m(-1)>-4(-1)\\ 6>m>4\end{array}$

This inequality can be written as:

$4<m<6$

The value of m lies between 4 and 6, where 4 and 6 are excluded.

Thus, the required solution of the given inequality is $4<m<6$ or $\left(4,6\right)$.