University:
Carleton College
Course:
MATH 111 | lntroduction to Calculus
Academic year:

2024
Views:

194

Pages:

3
Author:

Alecai8y

Question 1 Determine the absolute value of the complex number z = 5 + 12i Answer Step 1 We must determine the complex number's absolute value. z = 5 + 12i We know that if we have complex number z = x + iy then its absolute value will be, |z| = √(x² + y²) The complex number comparison z = 5 + 12i with z = x + iy, we get x = 5 y = 12 Where, x is real part of complex number and y is the imaginary part of the complex number. Step 2 Applying above formula for the given complex number, |z| = √(x² + y²) = √(5² + 12²) = √25 + 144 = √169 = 13 Hence, absolute value of complex number is 13. Question 2 A phrase that describes a set of real number is given. Express the phrase as an inequality involving an absolute value. All real numbers x at least 5 units from 7 Answer Step 1 All real numbers x at least 5 units from 7 that is, x ≥ 7 + 5 ⇒ x - 7 ≥ 5 x ≤ 7 - 5 ⇒ -(x - 7) ≥ 5 Step 2 We know that |u| ≥ a ⇒ u ≥ a or u ≤ -a...(3) Since we have, x ≥ 7 + 5 ⇒ x - 7 ≥ 5...(1) x ≤ 7 - 5 ⇒ x - 7 ≤ -5...(2) So using (1), (2) and (3) we can write, |x - 7| ≥ 5 Step 3 Hence |x - 7| ≥ 5 Question 3 The absolute value of -6 is equal to the distance on a number line between. 1)-6 an 6 2)-6 and 1 3)0 and 6 4)1 and -6 Answer Step 1 The absolute value of a number is the distance from 0 on number line. The absolute value of -6 is 6 because distance of -6 from 0 is 6. Absolute value measures how far is given number from zero. Now, from given options we have to check which option has same distance as absolute -6. option 1st is -6 and 6. Distance between -6 and 6 = |6 - (-6)| = 12 which is not equal to absolute value of -6 Therefore, this option is not correct. Option 2nd is -6 and 1. Distance between -6 and 1 = |1 - (-6)| = 7 which is not equal to absolute value of -6. Therefore, this option is not correct. Option 3rd is 0 and 6. Distance between 0 and 6 = |6 - 0| = 6 which is equal to absolute value of -6. Therefore, the absolute value of -6 is equal to the distance on a number line between 0 and 6. Step 2 Now, distance between 1 and -6 = |1 - (-6)| = 7. Therefore, this option is also not correct. Hence, option 3rd is correct. Question 4 To evaluate: The simplified form of the expression 1-0.06. Answer The given expression is 1 - 0.06. Write the subtraction in terms of addition with the use of definition that is, a-b=a+(-b) The given expression becomes, 1-0.06 = 1 + (-0.06) Now, the given two values are of different signs and for the addition of any two numbers of different signs, subtract the smaller absolute value from the larger absolute value. Absolute values of the numbers are, |1|=1 |-0.06|=0.06 The number 1 has larger absolute value so subtract 0.06 from 1. 1-0.06 = 0.94 As the largest number 1 is positive, the answer is also positive So, the difference of the given expression is 0.94. Final statement: Therefore, the simplified form of the expression 1-0.06 is 0.94. Question 5 Solve the absolute value equations for x. |2x-2|=|3x-6| Answer |2x-2|=|3x-6| First case: 2x-2=3x-6 3x-2x=6-2 x=4 Second case: 2x-2=-(3x-6) 2x-2=-3x+6 5x=8 x = 8/5