Q53. - Algebra 2

Question

Graph each set on a number line.

     a.$\mathbf{-}\mathbf{2}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{4}$

     b.$\mathbf{x}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{or}\mathbf{ }\mathbf{x}\mathbf{>}\mathbf{3}$

     c.$\left(\mathbf{-}\mathbf{2}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{4}\right)\mathbf{and}\left(\mathbf{x}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{orx}\mathbf{>}\mathbf{3}\right)$ (Hint: This is the intersection of the graphs in part a and part b.)

   d.Solve $\mathbf{3}\mathbf{<}\left|\mathbf{x}\mathbf{+}\mathbf{2}\right|\le \mathbf{8}$. Explain your reasoning and graph the solution set.

Step-by-Step Solution

Answer

a. The graph of the given inequality $\mathbf{-}\mathbf{2}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{4}$ on the number line is:

b. The graph of the given inequality $\mathbf{x}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{orx}\mathbf{>}\mathbf{3}$ on the number line is:

 c. The graph of the given inequality $\left(\mathbf{-}\mathbf{2}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{4}\right)\mathbf{and}\left(\mathbf{x}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{orx}\mathbf{>}\mathbf{3}\right)$ on the number line is:

d. The solution of the given inequality $\mathbf{3}\mathbf{<}\left|\mathbf{x}\mathbf{+}\mathbf{2}\right|\le \mathbf{8}$ is $\mathbf{-}\mathbf{10}\le \mathbf{x}\mathbf{<}\mathbf{-}\mathbf{5}\mathbf{or}\mathbf{1}\mathbf{<}\mathbf{x}\le \mathbf{6}$.

The graph of the given inequality $\mathbf{-}\mathbf{10}\le \mathbf{x}\mathbf{<}\mathbf{-}\mathbf{5}\mathbf{or}\mathbf{1}\mathbf{<}\mathbf{x}\le \mathbf{6}$ on the number line is:

 

a. Step 1 &shy;- Description of step.

The given inequality is: $-2<x<4$.

Step 2 &shy;- Description of step.

The given inequality $-2<x<4$ implies that $x$ is greater than $-2$ but less than 4.

Here inequality is not having equal to sign, therefore the values $-2$ and 4 are excluded.

Step 3 &shy;- Description of step.

The graph of the given inequality $-2<x<4$ on the number line is:

b. Step 1 &shy;- Description of step.

The given inequality is: $x<-1or x>3$.

Step 2 &shy;- Description of step.

The given inequality $x<-1orx>3$ implies that $x$ is less than $-1$ and also $x$ is greater than 3. 

Here inequality is not having equal to sign, therefore the values $-1$ and 3 are excluded.

Step 3 &shy;- Description of step.

The graph of the given inequality $x<-1orx>3$ on the number line is:

c. Step 1 &shy;- Description of step.

The given inequality is: $\left(-2<x<4\right)and\left(x<-1orx>3\right)$.

Step 2 &shy;- Description of step.

The solution of the given inequality $\left(-2<x<4\right)and\left(x<-1orx>3\right)$ will be the intersection of the graphs of the inequalities $-2<x<4$ and $x<-1orx>3$.

Step 3 &shy;- Description of step.

The graph of the given inequality $\left(-2<x<4\right)and\left(x<-1orx>3\right)$ on the number line is:

d. Step 1 &shy;- Description of step.

The given inequality is: $3<\left|x+2\right|\le 8$.

The solution of the given inequality $3<\left|x+2\right|\le 8$ can be find out by finding the intersection of the solutions of the inequalities $\left|x+2\right|\le 8$ and $\left|x+2\right|>3$.

Step 2 &shy;- Description of step.

As, it is known that if $\left|a\right|\le b$, then $-b\le a\le b$.

Therefore, the solution of the inequality $\left|x+2\right|\le 8$ is given by:

$\begin{array}{c}\mathbf{-}\mathbf{8}\mathbf{\le }\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{\le }\mathbf{8}\\ \mathbf{-}\mathbf{8}\mathbf{-}\mathbf{2}\mathbf{\le }\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{-}\mathbf{2}\mathbf{\le }\mathbf{8}\mathbf{-}\mathbf{2}\\ \mathbf{-}\mathbf{10}\mathbf{\le }\mathbf{x}\mathbf{\le }\mathbf{6}\end{array}$

Therefore, the solution of the inequality $\left|x+2\right|\le 8$ is $-10\le x\le 6$.

As, it is known that if $\left|a\right|>b$, then $a>b\text{ or a}<-b$.

Therefore, the solution of the inequality $\left|x+2\right|>3$ is given by:

$\begin{array}{c}x+2>3\text{ or }x+2<-3\\ x+2-2>3-2\text{ or }x+2-2<-3-2\\ x>1\text{ or }x<-5\end{array}$

Therefore, the solution of the inequality $\left|x+2\right|>3$ is $x>1\text{ or }x<-5$.

Step 3 &shy;- Description of step.

As, the solution of the given inequality $3<\left|x+2\right|\le 8$ can be find out by finding the intersection of the solutions of the inequalities $\left|x+2\right|\le 8$ and $\left|x+2\right|>3$.

Therefore, the solution of the given inequality $3<\left|x+2\right|\le 8$ is given by:

$\left(-10\le x\le 6\right)\cap \left(x>1\text{ or }x<-5\right)=-10\le x<-5\text{ or 1}<x\le 6$

Therefore, the solution of the given inequality $3<\left|x+2\right|\le 8$ is $-10\le x<-5\text{ or 1}<x\le 6$.

Step 4 &shy;- Description of step.

The graph of the given inequality $-10\le x<-5\text{ or 1}<x\le 6$ on the number line is: