# 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.