Brought To You By- Tutorial Services-The Math Center INEQUALITIES Brought To You By- Tutorial Services-The Math Center

Inequalities An inequality is a comparison of values There are four possible relationships between different values. Greater than, less than, greater than or equal to, and less than or equal to. The four symbols used to represent these relationships respectively are : >,<, ≥, ≤

In this case, the inequality states that 3 is less than 5. Example: 3 < 5 In this case, the inequality states that 3 is less than 5. Think about the symbol as being an arrow that always points toward the smaller value. 2 < 5 < 7 or 9 > 5 > 3

Inequalities also work with variables Example: x < 5 means that x can be any value that is less than 5. x ≥ 3 means that x is any number greater than or equal to 3.

Graphing Inequalities in One Variable The solution set for inequalities of one variable can be graphed on a number line. Example: x < 5 would be graphed as : ) If x is less than or greater than 5, a parenthesis or an open circle is used in the graph because 5 is not included in the set. x 5 -5 -OR- x 5 -5

[ Example: You would graph the solution of 3 ≤ x as follows x 5 3 x 5 In this case, a bracket or a closed circle is used because 3 is included in the set. x 5 3 [ x 5 3 -OR-

( ( [ ) x > 9 x ∞ 12 < x x ∞ 3 ≤ x < 5 x Some examples of graphing solutions x ∞ 9 x > 9 ( x ∞ 12 12 < x ( x 5 3 3 ≤ x < 5 [ )

Interval Notation Example: The solution set of 3 ≤ x < 5 Another way of representing the solution set of inequalities is by using interval notation. Example: The solution set of 3 ≤ x < 5 can be written using interval notation as [3,5). From this notation you can see that there is a left boundary and a right boundary which are 3 and 5. However, if the number is in parenthesis, then it is not in the set but, numbers close to that boundary, such as 4.9 or 4.99, are included.

What happens if one of the boundaries is removed? Example: x < 5 In this case, the values of x go on forever in the negative direction. We can represent this inequality in interval notation as (-∞,5), where the symbol (-∞), meaning negative infinity, is used to represent the fact that there is no endpoint for the values of X in the negative direction. Is represented by a graph as : x 5 -5 )

Examples of interval notation: x ≥ 3 Answer: [3,∞) Answer: (-∞,15 ) Answer: (3,15)

Solving Linear Inequalities of One Variable Example: x+3 < 4 is a linear inequality All of the rules that apply to equations also apply to inequalities with one important exception. If you ever multiply or divide by a negative number you must reverse the direction of the inequality sign.

Subtract 4 from both sides: -2x+4 -4 >8 -4 Example: -2x+4 > 8 To solve for x : Subtract 4 from both sides: -2x+4 -4 >8 -4 Divide both sides by -2: -2x / -2 > 4 / -2 Since we are dividing by a negative we have to change the direction of the inequality: X < -2 Answer: x < -2 or (-∞, 2)

Example with X’s on both sides: 4x-6 > 2x+10 Add 6 to both sides: 4x -6 +6 > 2x+10 +6 Subtract 2x from both sides: 4x -2x > 2x+16 -2x Divide both sides by 2: 2x /2 > 16 /2 Answer: x>8 or (8,∞)

Subtract 3 from both sides: x+3 -3 > 4x+6 -3 Example: x+3 > 4x+6 Subtract 3 from both sides: x+3 -3 > 4x+6 -3 Subtract 4x from both sides: x -4x > 4x+3 -4x Divide both sides by -3: -3x / -3 > 3 / -3 Change the direction of the inequality since there is division by a negative number Answer: x < -1 or (-∞,-1)

Distributive Property: 5x -10 ≥ 9x -6x +12 Example: 5(x-2) ≥ 9x-3(2x-4) Distributive Property: 5x -10 ≥ 9x -6x +12 Add like terms: 5x-10 ≥ 3x+12 Add 10 to both sides: 5x-10 +10 ≥ 3x+12 +10 Subtract 3x from both sides: 5x -3x ≥ 3x +22 -3x Divide both sides by 2: 2x / 2 ≥ 22 / 2 Answer: x ≥ 11 or [11,∞)

When you solve an inequality the solutions can be written in interval notation or set builder notation. Set builder notation: and {x | 5 < x ≤ 9} This notation says that x is any value such that it satisfies the inequality in the notation. {x | x ≤ 5}

Solving Compound Inequalities Compound inequalities involve two inequalities of the same variable. Example: -3 < 2x+1 < 5 can be written as two inequalities -3 < 2x+1 and 2x+1 < 5 Each inequality is solved separately. Another method to solve the compound inequality is to apply properties of equations to all parts of the inequalities.

Can be solved as a whole as follows: Example: -3 < 2x+1 < 5 Can be solved as a whole as follows: Subtract 1 from all parts: -3 -1 < 2x +1 -1 < 5 -1 Divide all parts by 2: -4 / 2 < 2x / 2 < 4 / 2 Answer: -2 < x < 2 or (-2,2) or {x | -2 < x < 2}

Solving Absolute Value Inequalities Solving inequalities involving absolute values requires learning 2 new rules that will affect how the problem is solved. 1. When solving an inequality in which the absolute value expression is less than (<) or less than or equal to (≤) a standard expression, such as |x+3| ≤ 5, we must rewrite this as -5 ≤ x+3 ≤ 5 and solve. 2. When solving an inequality in which the absolute value expression is greater than (>) or greater than or equal to (≥) a standard expression, such as |2x-1| > 7, we must rewrite this as 2x-1 < -7 or 2x-1 > 7 and solve each inequality separately.

Subtract 3 from all parts: -5 -3 ≤ x +3 -3 ≤ 5 -3 Example: |x+3| ≤ 5 Rewrite: -5 ≤ x+3 ≤ 5 Subtract 3 from all parts: -5 -3 ≤ x +3 -3 ≤ 5 -3 Simplifying gives answer (Note the answer is given in three different ways) : -8 ≤ x ≤ 2 or [-8,2] or {x | -8 ≤ x ≤ 2}

Simplifying gives the answer: x < -3 or x > 4, (-∞,-3) U (4,∞) Example: |2x-1| > 7 Rewrite: 2x-1 < -7 or 2x-1 > 7 Add 1 to both sides of each inequality: 2x -1 +1 < -7 +1 or 2x-1 +1 > 7 +1 Divide both sides of each inequality by 2: 2x /2 < -6 /2 or 2x /2 > 8 /2 Simplifying gives the answer: x < -3 or x > 4, (-∞,-3) U (4,∞) or {x | x < -3 or x > 4}

Rewrite: -1 < -2x +5 < 1 Example: |-2x+5| < 1 Rewrite: -1 < -2x +5 < 1 Subtract 5 from all parts -1 -5 < -2x +5 -5 < 1 -5 Divide all parts by -2: -6 /-2 < -2x /-2 < -4 /-2 Change direction of inequality: 3 > x > 2 Answer: (2,3) or {x | 2 < x < 3}

Links and handouts Inequalities student handout Inequalities Quiz