Algebra Answers Note #1

Question 1 Let $H$ be a subgroup of $\mathbb{R}^*$, the group of nonzero real numbers under multiplication. If $\mathbb{R}^+ \subseteq H \subseteq \mathbb{R}^*$, prove that $H = \ mathbb{R}^+$ or $H = \mathbb{R}^*$. Answer First, we need to understand the concept of subgroups and their relationship with the group they are a part of. A subgroup of a group is a subset of the group that still forms a group under the same operation. In this case, the operation is multiplication and the group is $mathbb{R}^*$. This means that for a subgroup to be valid, it must satisfy the following criteria: 1. Closure: For any two elements $a, b in H$, the product $ab$ must also be in $H$. 2. Identity element: The subgroup must contain the identity element of the group, which in this case is 1. 3. Inverses: For any element $a in H$, its inverse $a^{-1}$ must also be in $H$. Now, let's take a closer look at the given conditions, $mathbb{R}^+ subseteq H subseteq mathbb{R}^*$. This means that all positive real numbers are in $H$ and $H$ is a subset of all nonzero real numbers. This immediately tells us that $H$ contains the identity element (1) and all inverses of positive real numbers, since they are also in $mathbb{R}^*$. Since $H$ is a subgroup of $mathbb{R}^*$, it must satisfy the criteria mentioned earlier. Now, the key here is to notice that every element in $mathbb{R}^*$ can be written as a positive real number multiplied by its inverse. For example, $a in mathbb{R}^*$ can be expressed as $a = x times x^{-1}$, where $x$ is a positive real number. This shows that every element in $mathbb{R}^*$ can be generated using the elements of $mathbb{R}^+$. Thus, if $H$ contains all positive real numbers, it must also contain all other elements of $mathbb{R}^*$, making $H = mathbb{R}^*$. Similarly, if $H$ only contains positive real numbers, it satisfies all the criteria of a subgroup and is equal to $mathbb{R}^+$. Hence, we can conclude that $H = mathbb{R}^+$ or $H = mathbb{R}^*$. Question 2 A truck rental company rents a truck for one day by charging $28.00 plus $1.15 per mile. a. Express the cost to rent the truck for one day, C, as a function ofthe number of miles driven, x b. What is the cost of renting the truck if the truck was driven 58 miles? Answer a. To express the cost to rent the truck for one day, C, as a function of the number of miles driven, x, we will use the formula C = 28 + 1.15x, where 28 represents the base cost and 1.15 represents the cost per mile. This formula follows the linear function form, y = mx + b, where m represents the slope (cost per mile) and b represents the y-intercept (base cost). b. To find the cost of renting the truck if it was driven 58 miles, we will substitute 58 for x in the function C = 28 + 1.15x. This gives us C = 28 + 1.15(58) = $91.70. Therefore, the cost of renting the truck for one day and driving 58 miles would be $91.70. Question 3 Luis is looking at two triangles with side lengths given algebraic expressions. PART B The area of the first rectangle is and the area of the second rectangle is x2 + 7x + 12. What is the sum of the areas of the two rectangles? A x2 + 24x B 3x2 + 7x +24 C x2 + 7x + 18 D 2x2 + 12x + 12 Answer As a high school student, I understand the importance of accuracy and attention to detail in solving mathematical problems. Therefore, I will carefully analyze the given question and provide a thorough answer. First of all, let us review the given information. Luis is looking at two triangles with side lengths given as algebraic expressions. The area of the first triangle is missing, while the area of the second triangle is given as x^2 + 7x + 12. To find the sum of the areas of the two triangles, we need to add their individual areas. Since the area of the first triangle is missing, we need to use our algebraic skills to determine it. The formula for finding the area of a triangle is 1/2 * base * height. In this case, the base and height are represented by algebraic expressions. We can consider the base as 2x and the height as (x+2). Therefore, the area of the first triangle can be expressed as 1/2 * 2x * (x+2) = x^2 + 2x. Now, we can add the areas of the two triangles to get the sum: (x^2 + 2x) + (x^2 + 7x + 12) = 2x^2 + 9x + 12. The correct answer is option B - 3x^2 + 7x + 24. For a step-by-step breakdown, we can use the FOIL method to multiply (x+2) and (x+6). This gives us x^2 + 2x + 6x + 12 = x^2 + 8x + 12. Now, we can add this to the previously calculated area of the first triangle (x^2 + 2x) to get the final expression: x^2 + 2x + x^2 + 8x + 12 = 2x^2 + 9x + 12. In case you have any further doubts or need practice exercises, I highly recommend checking out online algebra resources such as Khan Academy, MathisFun, or Mathplanet. These websites provide step-by-step explanations and practice problems to strengthen your algebra skills. In conclusion, solving this question required a thorough understanding of algebraic expressions and the formula for finding the area of a triangle. By revising and practicing these concepts, we can improve our problem-solving skills and excel in mathematics. Question 4 Divide. If there is a remainder, include it as a simplified fraction. (4z2+9z–9)÷(z+3) Answer To solve this problem, we can use long division or synthetic division. I will be demonstrating the solution using long division. Step 1: First, we need to arrange the terms in descending order of their exponents. This will give us 4z^2 + 9z - 9 for the dividend and z + 3 for the divisor. Step 2: Now, we need to divide the first term of the dividend (4z^2) by the first term of the divisor (z), which gives us 4z. This will be the first term of our quotient. Step 3: Next, we multiply the first term of our quotient (4z) by the divisor (z + 3) and write the result below the dividend. This will give us 4z^2 + 12z. Step 4: Now, we need to subtract the result from the previous step from the first two terms of the dividend. This will give us 4z^2 + 9z - (4z^2 + 12z) = -3z. We bring down the last term of the dividend (-9). Step 5: We divide -3z by the first term of the divisor (z), which gives us -3. This will be the second term of our quotient. Step 6: We multiply the second term of our quotient (-3) by the divisor (z + 3) and write the result below the previous result. This will give us -3z - 9. Step 7: We subtract this result from the remaining terms of the dividend (-9 - (-9) = 0). This means that our divisor is evenly divisible by the dividend. Therefore, the remainder is 0. Step 8: We bring down the next term of the dividend, which is 0, and place it beside the result of our previous step. This gives us 0z. Step 9: We repeat the process by dividing 0z by z, which gives us 0. This will be the third term of our quotient. Step 10: We multiply the third term of our quotient (0) by the divisor (z + 3) and write the result below the previous result. This will give us 0z + 0 = 0. Step 11: We subtract this result from the remaining terms of the dividend (0 - 0 = 0). Again, this means that our divisor is evenly divisible by the dividend with no remainder. Therefore, the final quotient is 4z - 3 and the remainder is 0. This can also be written as the simplified fraction (4z - 3)/1. Our final answer is (4z - 3). Question 5 -2(bz - 5) = 16 The value of x in terms of b is ? The value of x when b is 3 is ? Answer First, we must understand that the given equation is in the form of "ax + b = c", where a = -2 and b = -10 (since -2 multiplied by (bz - 5) gives us -2bz + 10 and c = 16. Therefore, our goal is to isolate the variable, x, on one side of the equation. To find the value of x in terms of b, we will start by simplifying the equation: -2(bz - 5) = 16 -2bz + 10 = 16 (using distributive property) -2bz = 6 (subtracting 10 from both sides) bz = -3 (dividing both sides by -2) Now, we have the value of x in terms of b, which is -3. This means that when the value of b is given, we can easily substitute it into the equation to find the value of x. To find the value of x when b is 3, we simply substitute 3 into the equation: -2(3x - 5)= 16 (substituting 3 for b) -6x + 10 = 16 (using distributive property) -6x = 6 (subtracting 10 from both sides) x = -1 (dividing both sides by -6) Therefore, when b is 3, the value of x is -1.