How can we express Inequalities?ENGLISH ALGEBRA GRAPHICAL

(Number Line)

X is greater than 4 X > 4

X is greater than or equal to 4

X ≥ 4

X is less than 4 X < 4

X is less than or equal to 4

X ≤ 4

Solving InequalitiesWe deal with inequalities in a way similar to how deal with equations with one exception. An inequality is maintained if we:1. Add to or subtract from both sides the same

amount. 2. Multiply or divide both sides by the same amount

IF THAT AMOUNT IS POSITIVE.3. If we multiply or divide both sides by the same

NEGATIVE AMOUNT, THEN THE DIRECTION OF THE INEQUALITY IS REVERSED.

1. Solve and Graph• -3(2x-5)+1 ≥ 4

2. Word Problem

• A movie rental company offers two subscription plans. You can pay $36 per month and rent as many movies as you like, or pay $15 per month and pay $1.50 to rent each movie.

• How many movies must you rent in a month for the first plan to be cheaper than the second plan?

Can all inequalities be solved?• Are the following inequalities always true, never

true, or sometimes true?1. -2(3x+1) > -6x+7

2. 5(2x-3)-7x ≤ 3x+8

3. 4(2x-3) < 8(x-3)

Compound Inequalities

• Two types: “AND” and “OR”1. Find and graph the solution of: 7 < 2x+1 AND

3x ≤ 18. (This means that x must satisfy BOTH inequalities).

2. Find and graph the solution of: 7+k ≥ 6 OR 8+k < 3. (This means that k must satisfy EITHER inequality.)

Practice Problems• Textbook Pages 39-40, Do:51, 52, 53 (In this problem, find a relationship between AB and an actual number), and 67

Absolute Value Equations and Inequalities

• The absolute value of a real number is its distance from zero on a number on a number line. OR: the value of a number without regard to its sign.

• Algebraically: |x| = x, if x>0 |x| = -x, if x<0• So |x| is always positive.• Ex: If |x| = 4, then x=4 OR x=-4, since: |-4|=|4|=4

Extraneous Solutions

• Solve: |3x + 2| = 4x + 5• Do both solutions satisfy the original equation?• Sometimes, in the process of solving or simplifying

an equation we actually introduce solutions that don’t satisfy the original equation.

How can we solve equations with absolute values?

• Solve and Graph: |2x-1|= 5• Step 1: Rewrite the equation as two equations

without absolute values. Note that what is inside the | | can equal 5 or -5.

• Step 2: Solve each equation and Graph.• Step 3: Always check all of your solutions.• Try this one: 3|x+2|-1 = 8 (Hint: Get the absolute value alone before rewriting equations as two equations.)

How can we solve inequalities involving absolute value?

1. Write |x| < 5 as a compound inequality and graph the result.

2. Solve and graph |2x-1| < 53. Solve and graph |2x+4| ≥ 6