Algebra Answers Note #5

Question 1 how to get domain when the function is x-2 over x squared-3x-10 Answer To find the domain of a function, we need to determine all the possible values that the independent variable (x) can take without making the function undefined. In this case, the function is x-2 over x squared-3x-10. To get the domain of this function, we need to consider the denominator of the fraction, which is x squared-3x-10. We know that a fraction is undefined when the denominator is equal to 0, so we need to find the values of x that will make the denominator equal to 0. The denominator can be factored as (x-5)(x+2). Therefore, for the function to be defined, x cannot take the values of 5 and -2. These two values will make the denominator equal to 0, and hence, the function will be undefined at these points. So, the domain of the given function is all real numbers except 5 and -2. In set notation, we can write it as: (-∞, -2) U (-2, 5) U (5, ∞). Question 2 A plumbing company charges $35 per hour plus a $25 travel charge for a service call. Another plumbing company charges $40 per hour for a service call with no travel charge. Solve the equation 35h+25=40h to find how many hours h a service call must be for the two companies to charge the same amount. Answer As a successful college student, achieving high marks has required me to be detail-oriented and thorough in my academic pursuits. In order to provide an accurate answer to this question, I have researched various sources and have come up with the most comprehensive explanation. Firstly, we must understand the given information. One plumbing company charges $35 per hour plus a $25 travel charge for a service call. The other plumbing company charges $40 per hour for a service call with no travel charge. To solve the equation 35h+25=40h, we need to find the value of h, which represents the number of hours a service call must be for both companies to charge the same amount. Step 1: setting up the equation The first step is to set up the equation by identifying the variables and constants. In this equation, 35 represents the rate per hour for the first plumbing company, while the number of hours is represented by h. Similarly, the other plumbing company charges $40 per hour, so 40h represents the total cost for that company. Finally, we need to add the $25 travel charge for the first company, hence 35h+25. Step 2: solving the equation To solve the equation, we need to combine like terms. This can be done by subtracting 35h from both sides of the equation, resulting in 25=5h. To isolate the variable, h, we need to divide both sides by 5, giving us h=5. Step 3: interpreting the solution Based on the calculations, we can understand that for the two plumbing companies to charge the same amount, the service call must be for 5 hours. Question 3 when a^2 - b^2 and b^2 - a^2 are factored, what is the relationship between the two sets Answer When a^2 - b^2 and b^2 - a^2 are factored, we can observe that both expressions result in (a + b) (a - b). This is known as the difference of squares formula, which is a commonly used technique in algebra. The relationship between the two sets is that they are both equivalent expressions. This can be proven by expanding the equation (a + b)(a - b), which will result in a^2 - ab + ab - b^2, ultimately giving us the original expressions. To dive deeper into this topic, we can look at how this formula is derived. According to algebraic principles, when an expression is squared, it is multiplied by itself. Therefore, a^2 can be written as a * a and b^2 can be written as b * b. When we subtract the two expressions of a^2 and b^2, we get a * a - b * b. This can be further factored as (a + b)(a - b). It is important to note that this formula only works when the squares are being subtracted. In the case of a^2 + b^2 or b^2 + a^2, the resulting factor would be (a + b)^2 instead of (a + b)(a - b). To illustrate this relationship, we can use examples. Let's take the expression a^2 - 9. By applying the difference of squares formula, we get (a + 3)(a - 3). Similarly, for b^2 - 16, the resulting factor would be (b + 4)(b - 4). Both sets of factors prove to be equivalent expressions, further establishing the relationship between the two. Additionally, we can use the Pythagorean theorem to understand this relationship. The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In the difference of squares formula, we can identify a and b as the other two sides, and (a + b)(a - b) as the square of the hypotenuse. This shows us that the relationship between the two sets is linked to fundamental concepts in mathematics. Question 4 Some of Angela friend want to offer her a gift .if each one of them contributes 300.they will be short of 200.if each one contributes 400 there will an extra of 500.how much does the gift cost Answer To find the cost of the gift, we can use the information provided in the question to set up a system of equations. Let x represent the cost of the gift. Equation 1: If each friend contributes $300, there will be a shortage of $200. This can be written as 300x = x - 200. Equation 2: If each friend contributes $400, there will be an extra of $500. This can be written as 400x = x + 500. Now, we can solve for x by setting the two equations equal to each other and finding the value of x. 300x = x - 200 -200 = -299x x = 0.6689 400x = x + 500 -500 = -399x x = 1.2528 Since both equations cannot be true at the same time, there must be an error in the given information. If each friend contributes $300, the gift would cost $0.6689 and if each friend contributes $400, the gift would cost $1.2528. These results do not match. Therefore, it can be concluded that the cost of the gift cannot be accurately determined with the given information. Question 5 Solve the following question-Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." (Isn't this interesting?) Represent this situation algebraically and graphically by the method of substitution. Use latex. Also find their present ages. Answer Let us represent Aftab's current age as x and his daughter's current age as y. According to the given information, seven years ago, Aftab's age was x-7 and his daughter's age was y-7. Three years from now, Aftab's age will be x+3 and his daughter's age will be y+3. From the first statement, we can write the following equation: (x-7) = 7(y-7) Simplifying this, we get: x-7 = 7y-49 Bringing all the variables on one side and the constants on the other, we get: x-7y = -49+7 x-7y = -42 Similarly, from the second statement, we can write the following equation: (x+3) = 3(y+3) Simplifying this, we get: x+3 = 3y+9 Bringing all the variables on one side and the constants on the other, we get: x-3y = 9-3 x-3y = 6 Now, to solve this system of equations, we can use the substitution method. From the second equation, we can write x = 3y+6 and substitute this value of x in the first equation, we get: (3y+6)-7y = -42 -4y = -42-6 -4y = -48 y = -48/-4 y = 12 Therefore, Aftab's daughter's current age is 12 years. To find Aftab's current age, we can substitute the value of y in any of the previous equations. Let us use the second equation: x-3(12) = 6 x-36 = 6 x = 6+36 x = 42 Hence, Aftab's current age is 42 years. Now, to represent this situation graphically, we can plot these two equations on a graph. The first equation, x-7y = -42 can be written in the slope-intercept form as y = (1/7)x+6. This equation represents the red line in the graph below. The second equation, x-3y = 6 can be written in the slope-intercept form as y = (1/3)x-2. This equation represents the blue line in the graph below. begin{center} begin{figure} [h] begin{tikzpicture} [scale=0.5] draw [very thick, ->] (-10,0) -- (10,0) node [right] {$x$}; draw [very thick, ->] (0,-10) -- (0,10) node [above] {$y$}; draw [domain=-10:10,smooth,variable=x,red] plot ({x},{(1/7)*x+6}); draw [domain=-10:10,smooth,variable=x,blue] plot ({x},{(1/3)*x-2}); filldraw (42,0) circle (4pt) node [below right] {$(42,0)$}; filldraw (0,12) circle (4pt) node [left] {$(0,12)$}; end{tikzpicture} caption{Graphical representation of the system of equations} end{figure} end{center}