simplifying expressions in algebra lesson 1

mathematics

1. Lesson 1: Simplifying Expressions Prepared by: Hannaniah S. Jimanga

2. Objectives: At the end of this lesson, the learners are expected to: 1. Apply definitions, properties, axioms, theorems to simplify algebraic expressions, equations and inequalities.

3. Evaluate the following by applying the laws in operations of integers and grouping symbols. Show your step by step process.  

4. The Properties of the Real Numbers Commutative Property Commutative Property of Addition For any real number and , Commutative Property of Multiplication For any real number and ,  

5. The Properties of the Real Numbers Associative Property Associative Property of Addition For any real number , , Associative Property of Multiplication For any real number ,  

6. The Properties of the Real Numbers Distributive Property For any real number , ,  

7. The Properties of the Real Numbers Identity Properties Additive Identity Property For any real number , . Multiplicative Identity Property For any real number , .  

8. The Properties of the Real Numbers Inverse Properties Additive Inverse Property For any real number , there is a unique number such that . Note: The sum of a number a and its additive inverse is zero Multiplicative Inverse Property For any real number , there is a unique number such that . Note: The product of a number and its reciprocal is 1  

9. The Properties of the Real Numbers Example: Simplify the following expressions using the properties of real numbers.  

10. Lesson 2 Linear and Quadratic Equations

11. Equations An equation is a sentence that expresses the equality of two algebraic expressions. Given that , solve for the solution or root of the equation.  

12. The expression has ___ terms. ____ are the variable terms and ___ is the constant term. In each variable term, ____ is the numerical coefficient, and ___ is the variable part.  

13. Identify if the given values satisfies the equation (makes the equation true). when , and when , and when , , and  

14. Application The diameter of the base of a right circular cylinder is 5cm. The height of the cylinder is 8.5cm. Find the volume of the cylinder. Round to the nearest tenth.

15. Properties of Equality Addition Property of Equality (APE) Adding the same number to both sides of the equation does not change the solution set to the equation. In symbols, if , then . Multiplication Property of Equality Multiplying both sides of an equation by the same nonzero number does not change the solution set to the equation. In symbols, if and , then .  

16. Solving Equations Solve for the solution of the given equations by applying the different properties.  

17. Linear Inequalities A linear inequality in one variable is any inequality of the form , when and are real numbers, with . In place of we may also use .  

18. Properties of Inequalities Addition Property of Inequality If the same number is added to both sides of an inequality, then the solution set to the inequality is unchanged. Multiplication Property of Equality If both sides of an inequation are multiplied by the same positive number, then the solution set to the inequality is unchanged. If both sides of an inequation are multiplied by the same negative number, and the inequality symbol is reversed, then the solution set to the inequality is unchanged.

19. Strategy for Graphing a Linear Inequality Solve the inequality for y, and then graph . is satisfied above the line is satisfied on the line itself is satisfied below the line  

20. The Test-Point Method Strategy for Graphing a Linear Inequality by the Test Point Method Graph . Test any point not on the line to see if it satisfies the inequality. If the test point satisfies the inequality, shade the region containing the test point. If not, shade the other region.  

21. Quadratic Equations Quadratic equations has the form , where , , and are real numbers and . Zero Factor Property The equation is equivalent to the compound equation or .  

22. Solving Quadratic Equations by Factoring Example: Solve by factoring.  

23. Solving Quadratic Equations by Completing the Square,   Rule for finding the Last Term The last term of the a perfect square trinomial is the square of one-half of the coefficient of the middle term. In symbols, the perfect square whose first two terms are is  

24. Solving Quadratic Equations by Completing the Square,   Example: Solve the following by completing the square.  

25. Solving Quadratic Equations by Completing the Square,   Example:   Strategy: If , then divide each side by . Get only the and the terms on the left-hand side. Add to each side the square of the coefficient . Factor the left-hand side as the square of binomial. Solve for . Simplify.  

26. Solving Quadratic Equations by Quadratic Formula Th solution of with , is given by the formula:  

27. Number of solutions to a Quadratic Equation   The quadratic equation with has; Two real solutions One real solutions No real solutions (two imaginary solutions) Two real solutions One real solutions No real solutions (two imaginary solutions) discriminant

28. Number of solutions to a Quadratic Equation Example: Use the discriminant to identify the number of real solutions and find the roots by using the quadratic formula.  

29. Quadratic Inequality A quadratic inequality has one of the forms Where , , and are real numbers with .  

30. Lesson 3 Absolute Value Inequality

31. Absolute Value The absolute value of is a number whose distance from 0 on the number line is units. Example: Solution set:  

32. Absolute Inequality Basic Absolute Value Inequalities Absolute value inequality Equivalent Inequality Solution set Graph of the Solution Set Absolute value inequality Equivalent Inequality Solution set Graph of the Solution Set ) ( -k k ] [ -k k ( ) -k k [ ] -k k

33. Absolute Value Solve for Solution: No real numbers. Since , we write inequalities only when the value of is positive.  

34. Lesson 4 Mathematical modelling

35. Mathematical Modelling Mathematical modelling is the process of describing a real world problem in mathematical terms, usually in the form of equations, and then using these equations both to help understand the original problem, and also to discover new features about the problem. Formulate Solve Interpret Test

36. Mathematical Modelling Example: Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. What are the possible number of goals Alex scored? Formulate Solve Interpret Test

37. Mathematical Modelling Example: Regina makes $6.80 per hour in a café. To keep her scholarship grant, she may not earn more than $51 per week. What is the range of the number of hours per week that she may work? Formulate Solve Interpret Test

38. Mathematical Modelling Example: A boxing ring is in the shape of a square, 20ft on each sides. How far apart are the fighters when they are in opposite corners of the ring? Formulate Solve Interpret Test

39. Mathematical Modelling Example: Winston can mow his dad’s lawn in 1 hour less than it takes his brother Noel. If they take 2 hours to mow it when working together, then how long would it take Winston working alone? Formulate Solve Interpret Test