lect 22 quantitative reasoning.pptx .

Quantitative reasoning 1
Lecture number 22

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2. Quantitative Reasoning By Nazia Aslam Designation: Lecturer Niazi Medical & Dental College Sargodha 5/12/2024 2

3. Objectives Topic: Solution of linear and Quadratic Equation Definition of linear Equation Equation Solution Equivalent Equations Solution of a linear Equations Linear Equation with Fraction Rational Equation Quadratic Equation and Factoring Zero-Product Principle Solving a Quadratic Equation by Factoring 5/12/2024 3

4. A linear equation in one variable x is an equation that can be written in the form of: ax + b = 0 Where a and b are real numbers, and a An example of linear equation in one variable is: 4x + 12 = 0   5/12/2024 4 Definition of linear Equation

5. Equation Solution 5/12/2024 5 Solving an equation in x involves determining all values of ‘x’ that result in a true statement when substituted into the equation,. Such values are solutions, or roots, of the equation. For example, substitute -3 for ‘x’ in 4x + 12 = 0 4x + 12 = 0 4(-3) + 12 = 0 -12 + 12 = 0 0 = 0 This simplifies to the true statement 0 = 0, thus -3 is the solution of equation 4x + 12 = 0 The set of all such solution is called the equation solution .

6. Equivalent Equations Two or more equations that have the same solutions set are called equivalent equations . For example the equations 4x + 12 = 0 and 4x = - 12 if x = -3 than, These are equivalent equations because the solution set will be same for both equations. 5/12/2024 6

7. Solving a linear Equations The four steps are follows to solve the linear equation: Simplify the algebraic expressions on each side by removing grouping symbols and combining like terms. Collect all the variable terms on one side and all the numbers or constant terms, on the other side. Isolate the variable and solve. Check the proposed solutions in the original equation. 5/12/2024 7

8. Solve and check: 2(x-3) – 17 = 13 – 3(x+2) 5/12/2024 8 Step#01: simplifying the algebraic expression on each side. 2(x-3) – 17 = 13 – 3(x+2) 2x-6 – 17 = 13 – 3x-6 2x -23 = -3x + 7

9. 5/12/2024 9 Step#02: collect variable terms on one side and constant terms on the other side, we will collect variables of 2x – 23 = -3x + 7 on the left by adding 3x to both sides. We will collect number on the right by adding 23 to both sides. 2x – 23 + 3x = -3x +7 + 3x 5x - 23 = 7 5x – 23 + 23 = 7 + 23 5x = 30

10. 5/12/2024 10 Step#03: Isolate the variable and solve x by dividing both sides of 5x = 30 by 5. = x = 6  

11. 5/12/2024 11 Step#04: Check the proposed solution in the original equation, substituting 6 for x in the original equation. 2(6-3) – 17 = 13 – 3(6+2) 2(3) – 17 = 13 – 3(8) 6 – 17 = 13 – 24 -11= -11 the true statement verify that the solution set is {6}

12. Linear Equation with Fraction 5/12/2024 12 Solve and check: 12( .12 12( 4 3( 3x + 6 – 4x + 4 = 24 -x +10 = 24 - x = 24 -10 - x = 14 x = -14  

13. Rational Equation Solve: Solution: to identify the values of x that make denominators zero, lets factor the denominator on the right. This factorization is also necessary in identifying the least common denominator.   5/12/2024 13

14. 3(x - 2) + 1(x + 6) = 4 3x – 6 + x + 6 = 4 4x = 4 x = 1 Check the proposed solution. Substitute 1 for x in the original equation. You should obtain - - this true statement verifies that the solution set is {1}.   5/12/2024 14

15. Quadratic Equation and Factoring Linear equation are first degree polynomial equations of the form ax + b = 0. Quadratic equations are second degree polynomial equations and contain an additional term involving the square of the variable. Definition of a Quadratic Equation A quadratic equation in x is an equation that can be written in the general form of The a and be are real number and a , and it is also called second degree polynomial equation.   5/12/2024 15

16. The example of quadratic equation in general form: Some quadratic equations can be solve by factoring and using the zero-product principle.   5/12/2024 16

17. Zero-Product Principle If the product of two algebraic expressions is zero, than at least one of the factors is equal to zero. If AB = 0 then, A=0 or B=0 The zero-product principle can be applied only when a quadratic equation is in general form, with zero on one side of the equation. 5/12/2024 17

18. Solving a Quadratic Equation by Factoring If necessary, rewrite the equation in the general form , moving all terms to one side, thereby obtaining zero on the other side. Factor completely. Apply the zero-product principle setting each factor containing a variable equal to zero. Solve the equation in step 3. Check the solution in the original equation.   5/12/2024 18

19. Reference “Quantitative reasoning: Tools for Today informed Citizen” by Bernard L. Madison, Lynn and Arthur Steen. “Quantitative reasoning for the information Age” by Bernard L. Madison, David M. Bressoud . “Fundamentals of Mathematics” by Wade Ellis. 2/14/2024 19

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