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=cor 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|