Using Distance to Understand Absolute Value Equations and InequalitiesFor any two numbers...

Using Distance to Understand Absolute Value Equations and Inequalities

For any two numbers a and b on the number line, the distance between a and b can be written |a − b| or |b − a|. In exactly the same way, the equation |x − 3| = 4 can be read, “the distance between 3 and an unknown number is equal to 4.” The advantage of reading it in this way (instead of “the absolute value of x minus 3 is 4”), is that a much clearer visualization is formed, giving a constant reminder there are two solutions. In diagram form we have Figure 2.51.

Figure 2.51

From this we note the solutions are x = − 1 and x = 7.

In the case of an inequality such as |x + 2| ≤ 3, we rewrite the inequality as |x − (−2)| ≤ 3 and read it, “the distance between −2 and an unknown number is less than or equal to 3.” With some practice, visualizing this relationship mentally enables a quick statement of the solution: x ∈ [ − 5, 1]. In diagram form we have Figure 2.52.

Figure 2.52

Equations and inequalities where the coefficient of x is not 1 still lend themselves to this form of conceptual understanding. For |2x − 1| ≥ 3 we read, “the distance between 1 and twice an unknown number is greater than or equal to 3.” On the number line (Figure 2.53), the number 3 units to the right of 1 is 4, and the number 3 units to the left of 1 is −2.

Figure 2.53

For 2x ≤ − 2, x ≤ − 1, and for 2x ≥ 4, x ≥ 2, and the solution set is x ∈ (− ∞, −1] ∪ [2, ∞).

Attempt to solve the following equations and inequalities by visualizing a number line. Check all results algebraically.

|2x − 3| ≥ 5