# Topic 9 Absolute Value Equations

## 9.1 The Direction of a Number

Can you determine the value of the expression $$\dfrac{|x|}{x}$$ for all nonzero real number $$x$$ and explain the meaning of the value?

## 9.2 Properties of Absolute Values

The absolute value of a real number $$a$$, denoted by $$|a|$$, is the distance from $$0$$ to $$a$$ on the number line. In particular, $$|a|$$ is always greater than or equal to $$0$$, that is $$|a|\geq 0$$. Absolute values satisfy the following properties: $|-a|=|a|, \quad |ab|=|a||b| \quad \text{and} \quad \left|\frac{a}{b}\right|=\frac{|a|}{|b|}.$

An absolute value equation may be rewritten as $$|X|=c$$, where $$X$$ represents an algebraic expression.

If $$c$$ is positive, then the equation $$|X|=c$$ is equivalent to {$$X=c$$or $$X=-c$$.}

If $$c$$ is negative, then the solution set of $$|X|=c$$ is empty. An empty set is denoted by $$\emptyset$$.

More generally, $$|X|=|Y|$$ is equivalent to $$X=Y$$ or $$X=-Y$$.

The equation $$|X|=0$$ is equivalent to $$X=0$$.

Example 9.1 Solve the equation $|2x-3|=7.$

Solution.

The equation is equivalent to \begin{aligned} 2x-3 =-7 & \qquad\text{or} & 2x-3 =7 \\ 2x =-4 & & 2x =10 \\ x =-2 & \qquad\text{or} & x =5 \end{aligned}

The solutions are $$x=-2$$ or $$x=5$$. In set-builder notation, the solution set is $$\{-2, 5\}$$.

Example 9.2 Solve the equation $|2x-1|-3=8.$

Solution.

1. Rewrite the equation into $$|X|=c$$ form. $|2x-1|=11$

2. Solve the equation. \begin{aligned} 2x-1 =-11 & \qquad\text{or} & 2x-1 =11 \\ 2x =-10 & & 2x =12 \\ x =-5 & \qquad\text{or} & x =6 \end{aligned}

The solutions are $$x=-5$$ or $$x=6$$. In set-builder notation, the solution set is $$\{-5, 6\}$$.

Example 9.3 Solve the equation $3|2x-5|=9.$

Solution.

1. Rewrite the equation into $$|X|=c$$ form. $|2x-5|=3$

2. Solve the equation. \begin{aligned} 2x-5 =-3 & \qquad\text{or} & 2x-5 =3 \\ 2x =2 & & 2x =8 \\ x =1 & \qquad\text{or} & x =4 \end{aligned}

The solutions are $$x=1$$ or $$x=4$$. In set-builder notation, the solution set is $$\{1, 4\}$$.

Example 9.4 Solve the equation $2|1-2x|-3=7.$

Solution.

1. Rewrite the equation into $$|X|=c$$ form. $|2x-1|=5$

2. Solve the equation. \begin{aligned} 2x-1 =-5 & \qquad\text{ or } & 2x-1 =5 \\ 2x =-4 & & 2x =6 \\ x =-2 & \qquad\text{or} & x =3 \end{aligned}

The solutions are $$x=-2$$ or $$x=3$$. In set-builder notation, the solution set is $$\{-2, 3\}$$.

Example 9.5 Solve the equation $|3x-2|=|x+2|.$

Solution.

Note that two numbers have the same absolute value only if they are the same or opposite to each other. Then the equation is equivalent to
$3x-2=x+2\quad \text{or} \quad 3x-2=-(x+2).$ \begin{aligned} 3x-2 =x+2 & \qquad\text{or} & 3x-2 =-(x+2) \\ 2x =4 & & 4x =0 \\ x =2 & \qquad\text{or} & x =0 \end{aligned}

The solutions are $$x=2$$ and $$x=0$$. In set-builder notation, the solution set is $$\{0, 2\}$$.

Example 9.6 Solve the equation $2|1-x|=|2x+10|.$

Solution.

Since $$2$$ is positive, $$2|1-x|=|2||1-x|=|2-2x|$$. Moreover, because $$|-X|=|X|$$, the equation is equivalent to $|2x-2|=|2x+10|.$ \begin{aligned} 2x-2 =2x+10 & \qquad\text{or} & 2x-2 =-(2x+10) \\ -2 =10 & \qquad\text{or} & 4x =-8 \\ & & x =-2 \end{aligned}

The original equation only has one solution $$x=-2$$. In set-builder notation, the solution set is $$\{-2\}$$.

## 9.3 Practice

Problem 9.1 Find the solution set for the equation.

1. $$|2x-1|=5$$
2. $$\left|\dfrac{3x-9}{2}\right|=3$$

Problem 9.2 Find the solution set for the equation.

1. $$|3x-6|+4=13$$
2. $$3|2x-5|=9$$

Problem 9.3 Find the solution set for the equation.

1. $$|5x-10|+6=6$$
2. $$-3|3x-11|=5$$

Problem 9.4 Find the solution set for the equation.

1. $$3|5x - 2|-4 = 8$$
2. $$-2|3x + 1| + 5= -3$$

Problem 9.5 Find the solution set for the equation.

1. $$|5x-12|=|3x-4|$$
2. $$|x-1|=-5|(2-x)-1|$$

Problem 9.6 Find the solution set for the equation.

1. $$|2x-1|=5-x$$
2. $$-2x=|x+3|$$