Inequalities
H1 Understanding Inequalities
- Inequalities express that one quantity is greater than or less than another. They are represented by symbols such as < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
- Just as with equations, the goal when solving inequalities is to find the range of values that make the inequality true.
- The solutions to inequalities are expressed as ranges of values like x > 3 (x is greater than 3), or intervals like 2 < x ≤ 4 (x is greater than 2 and less than or equal to 4).
H1 Solving Simple Inequalities
- You solve inequalities in the same way as equations, but with one important difference: When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality changes.
- For example, when solving -2x < 6, divide through by -2 to get x > -3.
H1 Dealing with Inequalities Between Two Values
- Sometimes, inequalities will have a lower and an upper bound. This is expressed as a < x < b (a is less than x and x is less than b), or a ≤ x ≤ b (a is less than or equal to x and x is less than or equal to b).
- To solve these inequalities, carry out operations to all three parts at once.
- For example, to solve 2 < x/3 ≤ 4, multiply all parts by 3 to get 6 < x ≤ 12.
H1 Solving Quadratic Inequalities
- When solving a quadratic inequality such as x^2 < 9, first solve it as though it were an equality: x^2 = 9 has solutions -3 and 3. These values split the number line into three sections: Less than -3, between -3 and 3, and greater than 3.
- Substitute a value from each part into the inequality. For example, -4 gives 16 which isn’t less than 9, 0 gives 0 which is less than 9, and 4 gives 16 which isn’t less than 9. Therefore the solution is -3 < x < 3.
H1 Compound Inequalities
- Compound inequalities are two inequalities connected by the words “and” or “or”. Solutions must satisfy both inequalities in an “and” compound inequality, and either inequality in an “or” compound inequality.
H1 Graphing Inequalities
- When graphing inequalities, a basic understanding of graphing equalities is needed. The key difference is that instead of having just a line or curve, the solutions cover a range of values. Typically, you will shade the region where the inequality is true.
- For a simple linear inequality like y < 2x + 1, first graph the line y = 2x + 1, then shade the region below this line because we want where the y values are less than the line.
- Dashed or dotted lines are used to indicate that the points on the line aren’t included in the solution for “less than” or “greater than” inequalities. Solid lines indicate the points on the line are included in the solution for “less than or equal to” or “greater than or equal to” inequalities.
Always double-check your answers by substituting test values back into the original inequality.