Key Concepts
- Inequalities are solved using the following properties.
Addition Property
Let, [latex]a,b,[/latex] and [latex]c[/latex] be real numbers.
If [latex]a<b[/latex] then [latex]a+c<b+c[/latex].
This statement also holds for [latex]a \leq b, a>b,[/latex] and [latex]a \geq b[/latex].
Multiplication Property
Let, [latex]a,b,[/latex] and [latex]c[/latex] be real numbers.
If [latex]a<b[/latex] and [latex]c>0[/latex] then [latex]a \cdot c<b \cdot c[/latex].
If [latex]a<b[/latex] and [latex]c<0[/latex] then [latex]a \cdot c>b \cdot c[/latex].
This statement also holds for [latex]a \leq b, a>b,[/latex] and [latex]a \geq b[/latex].
- We solve an inequality by isolating the variable, just like when solving an equation, but remember to reverse the inequality when multiplying or dividing by a negative number.
- A compound inequality involving or has a solution set made up of all real numbers which satisfy the first inequality or the second inequality, or both.
- The solution to an and compound inequality is all the solutions that the two inequalities have in common. This is where the two graphs overlap.
Glossary
compound inequality: two inequalities joined with the word and or the word or.