1. x > y: “x is greater than y.”
2. x < y: “x is less than y.”
3. x ≥ y: “x is greater than or equal to y.”
4. x ≤ y: “x is less than or equal to y.”

    Solving inequalities is exactly like solving equations except for one very important difference: when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality switches.

 

    Here are a few examples:

 

 

    Notice that in the last example, the inequality had to be reversed. Another way to express the solution is x ≥ –2. To help remember that multiplication or division by a negative number reverses the direction of the inequality, remember that if x > y, then –x < –y, just as 5 > 4 and –5 < –4. Intuitively, this idea makes sense, and it might help you remember this special rule of inequalities.

 

    Absolute Value and Inequalities

 

    When absolute values are included in inequalities, the solutions come in two varieties.

1. If the absolute value is less than a given quantity, then the solution is a single range, with a lower and an upper bound. For example,

• First, solve for the upper bound:

 

• Second, solve for the lower bound:

 

• Now, combine the two bounds into a range of values for x. –1 ≤ x ≤ 5 is the solution.

1. The other solution for an absolute value inequality involves two disjoint ranges: one whose lower bound is negative infinity and whose upper bound is a real number, and one whose lower bound is a real number and whose upper bound is infinity. This occurs when the absolute value is greater than a given quantity. For example,

• First, solve for the upper range:

 

• Then, solve for the lower range:

 

• Now combine the two ranges to form the solution, which is two disjoint ranges: –∞ < x < –²⁰⁄₃ or 4 < x < ∞.

    When working with absolute values, it is important to first isolate the expression within absolute value brackets. Then, and only then, should you solve separately for the cases in which the quantity is positive and negative.

 

 

    Inequalities are also used to express the range of values that a variable can take. a < x < b means that the value of x is greater than a and less than b. Consider the following word-problem example: