Solving Equations with Fractions

2 Example: Solve for a. The LCD is 4. Simplify. Add 2a to both sides. Divide both sides by 3. Check your answer in the original equation.

3 Solving Equations with Fractions Example: Solve for x. Distribute to remove parentheses. The LCD is 12. Simplify. Combine like terms. Add 9x and subtract 32 from each side. Divide both sides by 17.

4 Procedure to Solve Equations 1.Remove any parentheses. 2.If fractions exist, multiply all terms on both sides by the LCD of the fractions. 3.Combine like terms, if possible. 4.Add or subtract terms on both sides of the equation to get all terms with the variable on one side of the equation. 5.Add or subtract a constant value on both sides of the equation to get all terms not containing the variable on the other side of the equation. 6.Divide both sides of the equation by the coefficient of the variable. 7.Simplify the solution, if possible. 8.Check your solution. Procedure to Solve Equations 1.Remove any parentheses. 2.If fractions exist, multiply all terms on both sides by the LCD of the fractions. 3.Combine like terms, if possible. 4.Add or subtract terms on both sides of the equation to get all terms with the variable on one side of the equation. 5.Add or subtract a constant value on both sides of the equation to get all terms not containing the variable on the other side of the equation. 6.Divide both sides of the equation by the coefficient of the variable. 7.Simplify the solution, if possible. 8.Check your solution.

5 Equations with No Solution Example: Solve for y. – 1 + 5(y – 2) = 12y + 3 – 7y – 1 + 5y – 10 = 12y + 3 – 7y Distribute to remove the parentheses. 5y – 11 = 5y + 3 – 11 = 3 Combine like terms. Subtract 5y from both sides. False! This equation has no solution.

6 Equations with Infinite Solutions Example: Solve for z. 7(z + 4) – 10 = 3z z – 2 7z + 28 – 10 = 3z z – 2 Distribute to remove the parentheses. 7z + 18 = 7z + 18 Combine like terms. Always true. This equation has an infinite number of solutions. 18 = 18 Subtract 7z from both sides.

Translating English Phrases into Algebraic Expressions

8 Translating Phrases into Expressions  multiplied by of product of times  less than decreased by smaller than fewer than shorter than difference of + greater than increased by more than added to sum of Represented by the Symbol: The English Phrase: Continued.

9 Translating Phrases into Expressions ÷ divided by ratio of quotient of 33 triple 22 double Represented by the Symbol: The English Phrase: = is was has costs equals represents amount to

10 Translating Phrases into Expressions Example: Write each English phrase as an algebraic expression. a.) Twelve more than a number x + 12 b.) Six less than one-third of the sum of a number and four

11 “The necklace costs $45 more than the bracelet.” Comparing Quantities Let b + 45 = the cost of the necklace. Let b = the cost of the bracelet. $45 more than the bracelet An expression may be written for several quantities using the same variable.

12 During the summer, Nancy read twice as many books as Andy. Marty read five more books than Andy. Expressions for Several Quantities Example: Use the letter A to write an algebraic expression for each quantity. Let 2A = the number of books Nancy read. Let A = the number of books Andy read. Twice as many as Andy Let 5 + A = the number of books Marty read. Five more than Andy

13 Expressions for Several Quantities Example: Write an algebraic expression for each quantity. The first angle of a triangle is 20 degrees more than the third angle. The second angle is double the third angle. Let x + 20 = the first angle. Let x = the third angle. 20 degrees more than the third angle Let 2x = the second angle. Double the third angle x x x2x

Solving Inequalities in One Variable

15 Inequalities An inequality is a statement that describes how two numbers are related to one another. 2– – 1– 3– 4– 5 – 4 < – 1 “is less than” – 1 > – 4 “is greater than” Smaller numbers are on the left. 3 < 10 4 > – 3 – 2 < 4

16 Graphing an Inequality Any number that makes an inequality true is called a solution of the inequality. The set of all numbers that makes the inequality true is called the solution set x ≤ 3 A closed circle is used to show that the endpoint is included in the answer. The symbols  and  will use a closed circle.

17 Graphing an Inequality An open circle is used to show that the endpoint is NOT included in the answer. The symbols > and < will use an open circle x <  x  3

18 Translating English Phrases into Algebraic Statements Example: Translate each English statement into an inequality. a.) To achieve a passing grade in her class, Patty must get at least an 80% on her final. x ≥ 80 b.) Fewer than 200 students were able to attend the assembly. x < 200

19 Solving Inequalities Rules when Inequalities Remain the Same For all real numbers a, b, and c: 1.if a > b, then a + c > b + c. 2.if a > b, then a  c > b  c. 3.If a > b and c is a positive number (c > 0), then ac > bc. 4.If a > b and c is a positive number (c > 0), then This property holds for all of the inequality symbols:, , .

20 Solving Inequalities in One Variable Example: Solve the inequality and graph the solution. k + 24  19 k  – 5 Subtract 24 from both sides. 55 44 33 22 1

21 Solving Inequalities in One Variable Example: Solve the inequality and graph the solution. f + 5  f  2 f  f – 7 Since 0 is always greater than –7, the solution is all real numbers. (Any value we put in for f in the original statement will give us a true inequality.) Subtract 5 from both sides. Subtract f from both sides. 0  – 7 55 44 33 22 1

22 Solving Inequalities in One Variable Example: Solve the inequality and graph the solution. 2(z – 3) < 4z z – 6 < 4z + 10 – 8 – 8) – 16 < 2z – 6 < 2z + 10 Remove parentheses. Divide both sides by 2. Subtract 10 from both sides. Subtract 2z from both sides.  10 88 66 44 2

23 Rules when Inequalities Are Reversed For all real numbers a, b, and c: 1.If a > b and c is a negative number (c < 0), then ac < bc. 2.If a > b and c is a negative number (c < 0), then This property holds for all of the inequality symbols:, , . Solving Inequalities in One Variable

24 Solving Inequalities in One Variable Example: Solve the inequality and graph the solution. 2(x – 3) < 4x x – 6 < 4x + 10 x >  8 – 2x < 16 – 2x – 6 < 10 Remove parentheses. Divide both sides by  2. Add 6 to both sides. Subtract 4x from both sides. The inequality sign is reversed.  10 88 66 44 2