Equations and inequalities with absolute value

|x − 3| = 2 can be read in two ways: “the absolute value of x − 3 is 2” or “on the number line, the point x is 2 units away from 3”. The second reading gives the answer at once: x = 1 or x = 5. We met |x| = a in “Positive and negative numbers, the number line and absolute value” and |x − a| < b in “Linear inequalities and systems”. DİM asks harder tasks: a quadratic inside the absolute value, an expression with x on the right, two absolute values, a parameter. The answer is often asked as “the sum of the roots” or “the number of integer solutions”.

Absolute value as a distance. |f(x)| = a

“Opening” an absolute value means looking at the sign of the expression inside: if it is not negative, simply drop the bars; if it is negative, write the expression with the opposite sign. For example, for x < 3, |x − 3| = −(x − 3) = 3 − x. Everything in equations, inequalities and simplifications rests on this rule. A few properties shorten the work:

|f(x)| = g(x) and |f(x)| = |g(x)|

Why is this condition important? The equations f = ±g are written without the absolute value, and among their roots there may be some with g(x) < 0 — at such a root the left side (a distance) is positive while the right side is negative. There is another way too: split into the cases f(x) ≥ 0 and f(x) < 0 and open the absolute value in each. If the inside is simple, the second way is shorter; if the right side is simple, the first one is. Whichever you choose, check the roots in the original equation at the end.

Several absolute values: the interval method

1. 1
   1. Zeros

   Set the inside of each absolute value to zero: x = 1 for |x − 1|, x = 5 for |x − 5|.

2. 2
   2. Intervals

   The zeros cut the number line into intervals; on each interval the insides keep their signs.

3. 3
   3. Open the absolute values

   Write an absolute value with a positive inside as it is, one with a negative inside with the opposite sign.

4. 4
   4. Solve and check

   Solve on each interval and check that the root lies in that interval; drop it if it does not.

5. 5
   5. Combine

   Write the roots from all intervals together.

Inequalities with absolute value

The language of distance helps here too: |x − a| < b means the points closer to a than b (an interval centred at a), |x − a| > b means the points farther than b (two rays). |x − a| < |x − b| means “x is closer to a than to b”: all points on a’s side of the midpoint of a and b.

For solving quadratic inequalities see “Quadratic inequalities and the interval method”. The interval method works for inequalities too: open the absolute values on each interval, solve the resulting inequality and intersect the result with that interval.

Graphs and a parameter: y = |f(x)| and y = f(|x|)

To draw y = |f(x)|, reflect the part of y = f(x) below the x-axis upwards. For y = f(|x|), keep the part of the graph with x ≥ 0 and reflect it to the left in the y-axis — the function becomes even (graph transformations: “Functions and their properties”). The number of roots of F(x) = a is the number of points where the graph y = F(x) meets the horizontal line y = a. In a parameter task we “slide” this line up and down.