INEQUALITIES

Introduction An inequality is similar to an equation except that the statement is that two expressions have a relationship other than equality, such as <, , > or .

Rules for inequalities Rule 1: An equal quantity may be added to (or subtracted from) both sides of an inequality without changing the inequality Example: Solve x – 3 > 5 Solution: x – 3 + 3 > 5 + 3 (add 3 to both sides) Thus, the solution is x > 8

Rules for inequalities (cont) Rule 2: An equal positive quantity may multiply (or divide) both sides of an inequality without changing the in equality Example: Solve 2x – 1 > 7 Solution: 2x – 1 + 1 > 7 + 1 (add 1 to both sides)  2x > 8  (divide both sides by 2)  x > 4 (the inequality remain the same)

Rules for inequalities (cont) Rule 3: If both sides of an inequality are multiplied (or divided) by a negative quantity then the inequality is reversed. Example: Solve 3 – 2x > -5 Solution: 3 – 3 – 2x > -5 – 3 (subtract 3 from both sides)  -2x < -8  (divide both sides by -2) (Apply rule 3, the inequality reversed)  x > 4

Solving linear inequalities Example 1: Solve the following inequalities. 7x – 6 > 3 + 2x 2 – 6x  4x + 3 3x < 9x + 4 4  3x – 2 < 13 x – 10 < 2x – 2 < x

Solving Quadratic Inequalities Quadratic inequalities need to handle with care. Example 2: Solve x2 – x – 12 > 0 Step 1 : Factorize the quadratic expression  (x + 3) (x – 4) > 0 Step 2: The critical points are: 4 and 3. Step 3 : Sketch a sign change diagram - + (x + 3) + (x – 4) - - + The solution is x < -3 or x > 4 + -3 - 4 +

Example 3: Solve 2x2  9x + 5. Step 1: Rearrange the inequality so that the expressions are on the left side and zero on the right side. 2x2 – 9x - 5  0 Step 2: Factorize the quadratic expression (2x + 1) (x – 5)  0 Step 3. The critical points are: 5 and -1/2 Step 4: Draw the sign change diagram The solution is -1/2  x  5 + - + -1/2 5

- + + Example 4: Solve (2 – x)(x + 3) > 0 Step 1: The critical points are: 2 and - 3 Step 2: Make sure the coefficient of x is positive.  (2 – x) = -x + 2 = -(x – 2) Hence (2 – x)(x + 3) >0 -(x – 2)(x + 3) > 0 Multiple both sides with -1,  -1  -(x – 2)(x + 3) < -1  0  (x – 2) (x + 3) < 0 Step 4: Draw a sign change diagram Note that the inequality is reversed The solution is -3 < x < 2 + - + -3 2

Solving Absolute Values Inequalities The absolute value of a real number x can be thought of as the distance from 0 to x on a real number line. |2| = 2 and |-2|= 2 Absolute value, denoted as |x|, is defined as follows |-x| |x| -x x

Solving Absolute Values Inequalities In a statement such as x < 2, it means that x must lie between 2 and -2. We can write this as |x|< 2  -2 < x < 2 Similarly, |x - 4|< 2 represents all points whose distance from point 4 is less than 2. |x - 4|< 2 -2 < x – 4 < 2  2 < x < 6 The points lie in the interval 2 < x < 6 < 2 < 2 x -x 4

Solving Absolute Values Inequalities If |x| > 5, this means the distance from 0 to x is greater than 5. This also indicates than the point x can be greater than 5 or less than -5. |x| > 5  x > 5 or x < -5 -x x -5 5

Solving Absolute Values Inequalities Properties : |x| = a if and only if x = ± a |x| < a if and only if –a < x < a |x|  a if and only if –a  x  a |x|  a if and only if x  a or x  - a |x| > a if and only if x > a or x < - a

Exercises: Solve the following inequalities