The Different Numbers

Simple Inequalities

A 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?

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.)

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. 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.)

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 inequalities involving absolute values? 1.Write |x| < 5 as a compound inequality and graph the result. 2.Solve and graph |2x-1| < 5 3.Solve and graph |2x+4| ≥ 6

More Absolute Value Practice

Complex Numbers

Graphically Represent and find the Absolute Value: i 6i 4-3i 3+4i

Algebra with Complex Numbers

Algebra with Complex Numbers (cont’d)

Reviewing Properties of Exponents Exponentiation is repeated Multiplication: To get to each succeeding line, divide by a: a 4 = a × a × a × a a 3 = a × a × a a 2 = a × a a 1 = a a 0 = 1 a -1 = 1/a a -2 = 1/a 2 a -3 = 1/a 3

Rules of Exponents

Try These

Roots and Radical Expressions

Some Exercises using Radicals TEXT: Page 365:27, 28, 35, 36, 37

Some Properties of Radicals

Express the Following in Simplest Form

How Can we Simplify this expression keeping with the Radical notation?

Binomial Radical Expressions

Simplify the Following Expressions

Simplify These

Another Notation: Rational Exponents

Examples

Solving Radical Equations  Method is similar to what we did with absolute value equations. Follow these Steps: 1)Isolate Radical; Rational exponent notation may be useful 2)Raise both sides to a power so that unknown will no longer be a radicand. 3)Raising radicals to powers can introduce extraneous solutions. Always check final answers in the original equation.

Solve the Following

Solving Equations with Two Radicals