Chapter 1, Problem 8CA

To determine

To identify: The solutions of each equation and fill that equation in one of the column.

Answer to Problem 8CA

The filled up table is represented below.

No solution	One solution	Two solutions	Infinitely many solutions
$0=\left|x+13\right|+2$$3x-12=3\left(x-4\right)+1$	$\left|8x+3\right|=0$$-6=5x-9$$-2x+4=2x+4$$12-2x=10x-8$	$9=3\left|2x-11\right|$	$7-2x=3-2\left(x-2\right)$$-4\left(x+4\right)=-4x-16$

Explanation of Solution

Given:

Total 9 equations are written below.

$\left|8x+3\right|=0$	$-6=5x-9$	$3x-12=3\left(x-4\right)+1$
$-2x+4=2x+4$	$0=\left|x+13\right|+2$	$-4\left(x+4\right)=-4x-16$
$12-2x=10x-8$	$9=3\left|2x-11\right|$	$7-2x=3-2\left(x-2\right)$

The first equation is $\left|8x+3\right|=0$ .

The modulus function is $\left|x\right|=\{\begin{array}{l}x\text{ if }x\ge 0\\ -x\text{ if }x<0\end{array}$ .

Use the modulus function to solve the equation $\left|8x+3\right|=0$ .

$\begin{array}{l}8x+3=0\text{ or }-\left(8x+3\right)=0\\ x=-\frac{8}{3}\text{ or }x=-\frac{8}{3}\end{array}$

In both the cases, there is only one solution for the equation $\left|8x+3\right|=0$ .

The second equation is $-6=5x-9$ .

Solve the equation $-6=5x-9$ for $x$ .

$\begin{array}{c}-6=5x-9\\ -6+9=5x\\ 3=5x\\ x=\frac{3}{5}\end{array}$

The equation $-6=5x-9$ has one solution.

The third equation is $3x-12=3\left(x-4\right)+1$ .

Solve the equation $3x-12=3\left(x-4\right)+1$ for $x$ .

$\begin{array}{c}3x-12=3\left(x-4\right)+1\\ 3x-12=3x-12+1\text{ distributive property}\\ -12=-11\end{array}$

The resultant equation can never be true

The equation $3x-12=3\left(x-4\right)+1$ has no solution.

The fourth equation is $-2x+4=2x+4$ .

Solve the equation $-2x+4=2x+4$ for $x$ .

$\begin{array}{c}-2x+4=2x+4\\ -2x-2x=4-4\\ -4x=0\\ x=0\end{array}$

The equation $-2x+4=2x+4$ has one solution.

The fifth equation is $0=\left|x+13\right|+2$ .

Solve the equation $0=\left|x+13\right|+2$ for $x$ .

$\begin{array}{c}0=\left|x+13\right|+2\\ \left|x+13\right|=-2\end{array}$

The absolute value cannot be negative

Therfore, the equation $0=\left|x+13\right|+2$ has no solution.

The sixth equation is $-4\left(x+4\right)=-4x-16$ .

Solve the equation $-4\left(x+4\right)=-4x-16$ for $x$ .

$\begin{array}{c}-4\left(x+4\right)=-4x-16\\ -4x-16=-4x-16\text{ distributive property}\\ -16=-16\end{array}$

The resultant equation is true for every real numbers.

Therfore, the equation $-4\left(x+4\right)=-4x-16$ has infinitely manysolutions.

The seventh equation is $-4\left(x+4\right)=-4x-16$ .

Solve the equation $12-2x=10x-8$ for $x$ .

$\begin{array}{c}12-2x=10x-8\\ 12+8=10x+2x\\ 20=12x\\ x=\frac{5}{3}\end{array}$

Therfore, the equation $12-2x=10x-8$ has one solution.

The eighth equation is $9=3\left|2x-11\right|$ .

The absolute function in the equation can be written as follows:

$\left|2x-11\right|=\{\begin{array}{l}2x-11\text{ if }x\ge \frac{11}{2}\\ -\left(2x-11\right)\text{ if }x<\frac{11}{2}\end{array}$

Substitute the value of the absolute function $\left|2x-11\right|$ for $x\ge \frac{11}{2}$ to solve the equation $9=3\left|2x-11\right|$ for $x$ .

$\begin{array}{l}9=3\left(2x-11\right)\\ 9=6x-33\text{ distributive property}\\ 6x=41\\ x=\frac{41}{6}\end{array}$

Substitute the value of the absolute function $\left|2x-11\right|$ for $x<\frac{11}{2}$ to solve the equation $9=3\left|2x-11\right|$ for $x$ .

$\begin{array}{l}9=-3\left(2x-11\right)\\ 9=-6x+33\text{ distributive property}\\ 6x=24\\ x=4\end{array}$

Therfore, the equation $9=3\left|2x-11\right|$ has two solutions.

The ninth equation is $7-2x=3-2\left(x-2\right)$ .

Solve the equation $7-2x=3-2\left(x-2\right)$ for $x$ .

$\begin{array}{c}7-2x=3-2\left(x-2\right)\\ 7-2x=3-2x+4\text{ distributive property}\\ 7-2x=-2x+7\\ 7=7\end{array}$

The resultant equation is true for all the real numbers.

Therfore, the equation $7-2x=3-2\left(x-2\right)$ has infinitely many solutions.

The filled up table is represented below.

No solution	One solution	Two solutions	Infinitely many solutions
$0=\left|x+13\right|+2$$3x-12=3\left(x-4\right)+1$	$\left|8x+3\right|=0$$-6=5x-9$$-2x+4=2x+4$$12-2x=10x-8$	$9=3\left|2x-11\right|$	$7-2x=3-2\left(x-2\right)$$-4\left(x+4\right)=-4x-16$