Equations, Inequalities, and Problem Solving
Inequalities
Solving an inequality is the process of finding the numbers that make an algebraic inequality a true numerical statement. We call such numbers the solutions of the inequality; the solutions satisfy the inequality.
There are various ways to display the solution set of an inequality. The three most common ways to show the solution set are set builder notation, a line graph of the solution, or interval notation.
Solving Inequalities
Addition Property of Inequality
Property
For all real numbers \(a\), \(b\), and \(c\)
$$ a > b \text{ if and only if } a + c > b + c $$This property is analogous for \(<\), \(\leq\) and \(\geq\).
Multiplication Property of Inequality
Property
For all real numbers \(a\), \(b\), and \(c\) with \(c > 0\)
$$ a > b \text{ if and only if } ac > bc $$Property
For all real numbers \(a\), \(b\), and \(c\) with \(c > 0\)
$$ a > b \text{ if and only if } ac < bc $$Equations and Inequalities Involving Absolute Value
Property 2.1
\(|x| = k\) is equivalent to \(x = - k\) or \(x = k\), where \(k\) is a positive number.
Solving Inequalities That Involve Absolute Value
Property 2.2
\(|x| < k\) is equivalent to \(x > - k\) or \(x < k\), where \(k\) is a positive number.
Remember that we can write a conjunction such as \(x > -k\) and \(x < -k\) in the compact form \(-k < x < k\).
Property 2.3
\(|x| > k\) is equivalent to \(x < - k\) or \(x > k\), where \(k\) is a positive number.