Question

How do you solve and write the following in interval notation:$-12<3-3x\le 15$?

Answer 1

Step-1: Separate the inequalities:

The given inequality is,

$\Rightarrow$$-12<3-3x\le 15$

In order to solve these simultaneous inequalities, need to separate as two inequalities.

$\Rightarrow$$\left\{\begin{array}{l}-12<3-3\mathrm{x}\\ 3-3\mathrm{x}\le 15\end{array}\right.$

Simple numerical terms are commonly written last.

$\Rightarrow$$\left\{\begin{array}{l}-12<-3\mathrm{x}+3\\ -3\mathrm{x}+3\le 15\end{array}\right.$

Step-2: Solve the expression:

In order to solve these simultaneous inequalities,need to group all the variable terms.

$\Rightarrow$$\left\{\begin{array}{l}-12+(-3)<(-3x+3)+(-3)\\ (-3x+3)+(-3)\le 15+(-3)\end{array}\right.$

Need to get rid of expression parentheses. In the given function there is no negative expressions

$\Rightarrow$$\left\{\begin{array}{l}\\ -3\mathrm{x}+3-3\le 15-3\\ -3\mathrm{x}\le 12\\ \frac{3\mathrm{x}}{3}\ge -\frac{12}{3}\end{array}\right.$

Numerical terms in these expressions are added.

Step-3:Satisfy the inequalities:

Solve the system of inequalities is equivalent to fixing all values of $\mathrm{x}$ that simultaneously satisfy the inequalities.

$\Rightarrow$$$\begin{gathered} -12+(-3)<(-3x+3)+(-3) \\ \left\{\begin{array}{l}\frac{3\mathrm{x}}{3}<\frac{15}{3}\\ \mathrm{x}\ge -4\end{array}\right. \\ \left\{\begin{array}{l}x<5\\ x\ge -4\end{array}\right. \end{gathered}$$
$\Rightarrow$$-4\le \mathrm{x}<5$
$\Rightarrow$$\mathrm{x}\in \left[-4,5\right]$

Hence, the system of inequalities is $\mathrm{x}\in \left[-4,5\right]$