12/5/17 Unit 4 Unit 4 - Equations and Inequalities - Vocabulary Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line
Review Vocabulary: Absolute value Additive inverse Coefficient Distributive Property Numerical Expression Algebraic Expression Integer Opposite Rational number Reciprocal
Equation A mathematical sentence that includes an equals sign to compare two expressions. Example: 2x + 1 = 9
Substitution To replace variables in an expression with numbers.
Evaluate Substitute a value for each variable, and then follow the order of operations. Example: evaluate: 2x + 1, when x = 3 2 3 + 1 = 7
Solve an Equation Apply inverse operations in order to isolate the variable. Example: x + 2 = 8 x + 2 2 = 8 2 x = 6
Complex Fraction A fraction where the numerator, denominator, or both contain a fraction. Example: 3 4 1 5 Main Fraction Bar
Solve a Complex Fraction Rewrite the complex fraction as a division expression. The main fraction bar becomes a division sign. 3 4 1 5 = 3 4 1 5 = 3 4 5 1 = 15 4
12/6/17 Unit 4 Unit 4 Solving One-Step Equations Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line
The Golden Rules of Equation Solving 1. Undo what has been done to the variable. 2. Always do the same thing to both sides of the equation.
To solve an equation you perform inverse operations. Operation Inverse Operation + +
Al Karaba Solving Equations 1 Addition & Subtraction (see video assigned in digits)
Example 1 x + 12 = 4 x + 12 12 = 4 12 x = 8
Example 2 7 = x 8 7 + 8 = x 8 + 8 15 = x
Al Karaba Solving Equations 2 Multiplication & Division (see video assigned in digits)
Example 3 5x = 35 5x 5 = 35 5 x = 7
Example 4 4 = x 3 4(3) = x 3 (3) 12 = x
12/11/17 Unit 4 Solving Two-Step Equations Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line
To simplify expressions use proper order of operations (GEMDAS). 7 + 4 5 7 + 20 27 Multiplication occurs before addition
To solve equations use inverse operations. Because we are undoing operations, this means doing GEMDAS backwards.
SADMEG I don t know how to solve two-step equations!
Everything You Need to Know About Solving Two-Step Equations To solve equations we undo addition and subtraction before we undo multiplication and division.
Al Karaba Solving Two-Step Equations Solving Two-Step Equations (see video assigned in digits)
Example 1: 5x + 7 = 32 7 7 5x = 25 5 5 Undo addition before undoing multiplication x = 5
Example 2: Undo subtraction x 3 8 = 5 before undoing division +8 + 8 x 3 = 3 3 3 x = 9
Example 3: 2x + 5 = 19 5 2x = 19 5 5 2x = 14 Undo addition before undoing multiplication 2 2 x = 7
12/12/17 Unit 4 Begin on a new page Solving Equations (if the variable is being multiplied by a fraction) Write the date and unit in the top corners of the page Write the title across the top line
To solve an equation in which the variable is being multiplied by a fraction: Multiply both sides of the equation by the reciprocal of the fraction.
Example 1: 5 2 2 5 x = 12 5 2 x = 60 2 = 30
Example 2: 11 4 4 11 x = 8 11 4 x = 88 4 = 22
Example 3: 3 5 x 1 = 8 +1 + 1 5 3 3 5 x = 9 5 3 x = 15 Undo subtraction before undoing multiplication by a fraction
1/2/18 Unit 4 Solving Equations with the Distributive Property Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line
How can I solve the following? 3 4 8x 4 = 1 Method 1: Use the Distributive Property Method 2: Use SADMEG
Example 1: Dist. Prop. 3 4 8x 4 = 1 3 4 8x 3 4 (4) = 1 6x 3 = 1 +3 + 3 6x = 4 6 6 x = 2 3
Example 1: SADMEG 4 3 3 4 8x 4 = 1 4 3 Operations on x: 1. 8 3. 8 2. 4 2. +4 3. 3 1. 3 or 4 4 4 3 8x 4 = 4 3 +4 + 4 8x = 16 3 8 8 x = 16 1 3 8 x = 2 3
Example 2: Dist. Prop. 1 2 3x 5 = 8 1 2 3x 1 2 (5) = 8 3 2 x 5 2 + 5 2 = 8 + 5 2 2 3 3 2 x = 21 2 x = 7 2 3
Example 2: SADMEG 2 1 1 2 3x 5 = 8 2 1 Operations on x: 1. 3 3. 3 2. 5 2. +5 3. 1 1. 1 or 2 2 2 3x 5 = 16 +5 + 5 3x = 21 3 3 x = 7
1/4/18 Unit 4 Solving Inequalities Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line
h 48
d 70%
a 12
2x 5 > 6
Inequality A mathematical sentence that uses <, >,,, or to compare the value of two expressions. Example: x < 8
Solution of an Inequality Any value of the variable that makes the inequality true. x < 8 x = 7.99, 6, 1 3, 567
Solve an Inequality Apply inverse operations in order to isolate the variable. Example: x 5 < 11 x 5 + 5 < 11 + 5 x < 16
Symbol Meaning < Less than Less than or equal to > Greater than Greater than or equal to Not equal to
Graphing Inequalities Inequality < > or or or Graphing Symbol Meaning The number graphed is NOT part of the solution The number graphed IS part of the solution
Example #1: x 2
Example #2: x + 7 < 11 7 7 x < 4
Example #3 x 5 6 +5 + 5 x 1
1/9/18 Unit 4 Solving Inequalities 2 Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line
Consider: 2 < 5 2 2 < 2 5 4 < 10
Consider: 2 < 5 ( 2) 2 < ( 2) 5 4 > 10
Consider: 3 < 9 3Τ 3 < Τ 1 < 3 9 3
Consider: 3 < 9 3Τ 3 < 9 Τ 3 1 > 3
When multiplying or dividing BOTH sides of an inequality by a NEGATIVE number, you must REVERSE the direction of the Inequality symbol.
Example #1: 3x < 12 3 3 x < 4
Example #2: 3x 12 3 3 x 4 Reverse the inequality symbol. BOTH sides divided by a negative!
1/4/18 Unit 4 Checking the Solution to an Inequality Begin on a new page Write the date and unit in the top corners of the page Write the title across the top line
Checking an equation: I solved 4x 11 = 17 and got x = 7. Is it correct? Substitute 7 for x and see if the equation is true: 4x 11 = 17 4 7 11 = 17 28 11 = 17 17 = 17 YES, my answer is correct
To check the solution to an inequality: 1. Check the related equation. 2. Check the direction of the Inequality
Example 1: Is x < 4 the solution of 4x + 3 < 19? Step 1: Write the inequality as an equation: 4x + 3 = 19 4(4) + 3 = 19 16 + 3 = 19 19 = 19 YES, the related equation is correct
Continued: Is x < 4 the solution of 4x + 3 < 19? Step 2: Pick a value from the solution and see if it works: pick any number less than 4 4x + 3 < 19 4(0) + 3 < 19 3 < 19 YES, solution is correct
Example 2: Is x 6 the solution of 3 x 1 5? 2 Step 1: Write the inequality as an equation: 3 3 2 x 1 = 5 2 6 1 = 5 9 1 = 5 8 = 5 NO, the related equation is incorrect Fix: Start over
Example 3: Is x 3 the solution of 5x + 2 17? Step 1: Write the inequality as an equation: 5x + 2 = 17 5( 3) + 2 = 17 15 + 2 = 17 17 = 17 YES, the related equation is correct
Continued: Is x 3 the solution of 5x + 2 17? Step 2: Pick a value from the solution and see if it works: pick any number less than 3 5x + 2 < 17 5( 5) + 2 < 17 25 + 2 < 17 27 < 17 NO, inequality is the wrong one! Fix: flip the inequality