Unit 1 Section 6 - 1.6 Equations and Inequalities Algebraic...

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1.6. Equations and Inequalities Algebraic expressions are merely symbolic forms of numbers (real or complex or their subsets); either, as counts (how many?) or as measures (how long?, how wide?, how thick?). For example, the algebraic expression r 2 h represents the volume of a right circular cylinder whose radius is r and whose height is h as seen in the figure on the right. Also, if you can do a piece of work in x days and your friend can do it in y days, then y x 1 1 represents the amount of work done with both you and your friend working together. Algebraic expressions can be compared with a constant or any other algebraic expression. This gives rise to equations and inequalities. An equation is a statement that two algebraic expressions are equal. An inequality is a statement that two algebraic expressions are not equal. For example, we may assert that the volume of a right circular cylinder is cu. units and its height is 3 times the radius, then we will arrive at the following equation, after substitution of the given values: r 2 (3r) = . As another example, we may want to find all numbers whose distance from 3 is not greater than 4. We write this as an inequality in this manner: |x – 3| 4 In any equation or inequality, it is necessary to identify which are the variables and which are constants and the relations that exist among them. Equations/inequalities naturally arise at this phase of problem solving. Time to think! A. Express the following quantities as algebraic expressions: 1. The greater part of 21, if the smaller part is x 2. The even integer consecutive to (and larger than) n , which is known to be even 3. The perimeter of a rectangle, two adjacent sides of which are a and b 4. The distance traveled by a man in t minutes, walking r miles per hour 5. Three consecutive odd integers, if x is the smallest 6. A number of two digits, if the unit’s digit is x and the ten’s digit is y 7. The new length of a piece of cloth after it is shrunk 1/5 of its original length, the original length being x 8. The hypotenuse of a right triangle if one leg is half of the other in measure B. If x is the velocity of a crew in still water, y is the velocity of the current and s is distance, what does the following represent? a) x + y b) x – y y x s ) d y x s ) c
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Unit 1.Algebra as the Study of Structures Section 6. page 2 The process of representing quantities as algebraic expressions and identifying the equation or inequality involved in a problem is called formulating the problem and the set of equations/inequalities and variables involved is called a formulation of the problem. Once a problem has been formulated, half of the task of problem solving is done. To solve an equation/inequality means to find the solution set , the set of all real or complex values of the variables that satisfy the given equation/inequality. Algebra provides the tools for solving these equations or inequalities. To make a systematic study of these tools, let us start
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Unit 1 Section 6 - 1.6 Equations and Inequalities Algebraic...

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