Questions and Answers #8 Equations and Inequalities

Questions and Answers Sheet 8 Equations and Inequalities Question #1 What is the constant term in the expansion of the binomial (2𝑥 + 3)3 ? Answer: During expansion (2𝑥 + 3)3 Each term consists of two parts. (a) "the powers of "2𝑥 " &𝑎𝑚𝑝; " 3 the respective terms will be term will have (2𝑥)3 ; (2𝑥)2 (3); (2𝑥)(3); (3)3 (b) the coefficients according to Pascals triangle, which will be 1,3,3,1 so looking at the two we see that the consta term will be 1 × 33 = 27 Question #2 How many solutions does the equation −4(𝑥 + 5) = −4𝑥 − 20 have? Answer: Increasing the bracket results in: −4𝑥 − 20 = −4𝑥 − 20 −4𝑥 + 4𝑥 = 20 − 20 0 = 0 Which is true After speaking with a friend, I've been told that the following is applicable: The outcome is always zero on both sides and thus always true. From this it means that despite this one outcome you can allocated any value to x. So there is a infinite number of solutions. Question #3 How to solve 9𝑥 − 7 = −7 ? Answer: Add 7 both sides: 9𝑥 − 7 = −7 9𝑥 − 7 + 7 = −7 + 7 9𝑥 = 0 𝑥 = 0/9 = 0 𝑥=0 Verify the response: 9(0) − 7 = 0 − 7 = −7 Question #4 How to solve 𝑦 = 𝑎𝑥 2 + 𝑐 for x? Answer: Assuming 𝑎! = 0 To begin, take c away from both sides to obtain: 𝑦 − 𝑐 = 𝑎𝑥 2 Transpose and divide both sides by a to get: 𝑥 2 = (𝑦 − 𝑐)/𝑎 So x must be a square root of (𝑦 − 𝑐)/𝑎 and we can deduce: 𝑥 = ±𝑠𝑞𝑟𝑡((𝑦 − 𝑐)/𝑎) Question #5 What is the value of T in the equation: 1/𝑇 − 1/200 = .010 ? Answer: We must solve for or isolate the variable T in order to determine its value. We accomplish this by applying the equation's given reverse operations. Knowing this, we can being to find T: 1/𝑇 − 1/200 = 0.010 1 1 𝑇 200 =≥; = 0.010 + Here, everything needs to be changed into a similar form, so we should now have: 1 1 1 =≥; = + 𝑇 100 200 Now we need common denominators: 1 2 1 =≥; = + 𝑇 200 200 Combine: 1 3 =≥; = 𝑇 100 All we have to do from here is cross-multiply: =≥; 3𝑇 = 100 Solve for T: =≥; 𝑇 = 100/3 Question #6 The sum of two numbers is 17. One number is 3 less than 2/3 of the other number. What is the lesser number? Answer: We can write the following by dialing these fictitious numbers x and y: 𝑥 + 𝑦 = 17 "One number is 3 less than 2/3 of the other number," we then see. Let's say that #y# is the "one number." If y is 3 less than 2/3 of x, this is written as: 𝑦 = 2/3𝑥 − 3 From here, we have to take this equation for y and use it in our first equation. Since we know that y and 2/3𝑥 − 3 are equal, we can replace y with 2/3𝑥 − 3 in the first equation: 𝑥 + 𝑦 = 17 𝑥 + (2/3𝑥 − 3) = 17 From here, we can add 𝑥 + 2/3𝑥 using fractions: 𝑥 + 2/3𝑥 = 3/3𝑥 + 2/3𝑥 = 5/3𝑥 5/3𝑥 − 3 = 17 Add 3 to both sides of the equation: 5/3𝑥 = 20 Multiply both sides by 3/5 : 𝑥 = 20 × 3/5 = (5(4)(3))/5 = 12 If 𝑥 = 12 , then 𝑦 = 5 , since their sum is 17. So the two numbers are 5 and 12. Question #7 How to solve for y in the equation 2𝑥 + 2𝑦 = 10 ? Answer: We must put all other terms on the right side and isolate the y term on the left in order to solve for y. Remember to do the reverse operation when moving terms across the '=' sign. Subtraction is the opposite of addition, and vice versa. The same is true for division and multiplication. hence : 2𝑦 = 10 − 2𝑥 y is being multiplied by 2 , so divide by 2 to obtain y. ⇒= (10 − 2𝑥)/2 = 10/2 − (2𝑥)/2 = 5 − 𝑥 Question #8 What is the slope of 𝑦 = 3/4𝑥 − 7 ? Answer: If a line's equation is similar to: 𝑦 = 𝑚𝑥 + 𝑏 The slope of that line is then m, and the point at which the line intersects the y-axis is b. Therefore, the slope is 3/4 .