Write inequalities corresponding to problem situations.
Write an interval using inequalities and graph it.
Solve linear inequalities.
Solve application problems involving linear inequalities.
A inequality is an algebraic expression involving less than (), less than or equal to (), greater than (), or greater than or equal to ().
Using Inequalities to Describe Intervals
Inequalities are often used to describe intervals. Here are some examples.
Conditions
Set Notation
Interval Notation
is greater than
is less than or equal to
is less than and greater than
is less than and greater than or equal to
is less than or is greater than
The last three examples are called compound inequalities. Compound inequalities are associated with the words "and" or "or". "And" inequalities describe intersections of sets, whereas "or" inequalities describe unions. We use the symbol to denote a union, and the symbol to denote an intersection.
To graph an inequality, we shade the interval(s) along a number line. Use parentheses ( ) and open dots to exclude endpoints. Use brackets [ ] and closed (filled-in) dots to include endpoints.
Examples
[1] Write an inequality that represents the following problem situation: The truck weighs less than 2000 lbs. (Solution)
[2] Write an inequality that represents the following problem situation: The car's fuel efficiency is no less than 40 mpg. (Solution)
[3] On a number line, graph the inequality . (Solution)
[4] On a number line, graph the inequality . (Solution)
[5] Write the algebraic inequality corresponding number line shown below. (Solution)
[6] On a number line, graph the compound inequality or . (Solution)
[7] On a number line, graph the compound inequality . (Solution)
[8] Write the algebraic inequality corresponding number line shown below. (Solution)
Solving inequalities
To solve an inequality means to find ALL replacements for the variable that make the inequality true. Your answers will typically (but not always) be intervals.
To solve a linear inequality, we apply the same process as we used for solving linear equations, with one exception:When solving a linear inequality, whenever you multiply (or divide) across by a negative number, you must reverse the inequality symbol(s).
Examples
[9] Find the solution set. Write the solution set in interval notation and graph it on a number line.