An inequality, similar to an equation, is a mathematical statement comparing two quantities. Some inequalities indicate that the two quantities are not necessarily equal, while others are strictly never equal.
|Symbol||Meaning||Strict / Non-strict|
|is less than||Strict|
|is less than or equal to||Non-strict|
|is greater than||Strict|
|is greater than or equal to||Non-strict|
When an inequality contains an unknown variable, it's possible to solve the inequality. A solution of an inequality is any value of the variable that makes the inequality true. For instance, the inequalityhas the solution because replacing with yields a true statement. Notice that is not the only value that solves and also work. In fact, most inequalities have an infinite number of solutions. The set of these solutions is called the solution set.
Is an element of the solution set of the following inequality?
If a value is a solution to an inequality, it is an element of the solution set. To determine if is a solution, we can substitute the value into the inequality and evaluate. If the resulting inequality is true, the value is a solution; if it's false, it is not a solution.
The graph of a one-variable inequality is a visual representation of the inequality's solution set, which can be drawn on a number line in three steps:
An arrow in either direction indicates that all numbers in that direction are part of the solution set.
Graph the inequality on a number line.
The inequality reads is less than which means that is not a solution to the inequality, but every value of less than is. Thus, our graph must have an open circle at
Every value of less than also has to be included in the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.
We have now fully graphed the inequality
Write the inequality shown by the graph.
To begin, let's use the variable in the inequality. From the graph, we can see a closed circle at Therefore, is a solution of the inequality. The arrow pointing toward the right indicates that all values greater than are also part of the solution set. Thus, the inequality would read is greater than or equal to which is expressed algebraically as