Maintaining Mathematical Proficiency

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Exercises 1 To graph 6 on the number line, place a point on 6.
Exercises 2 The absolute value of a number is the distance on a number line that number is from 0. Then, ∣2∣=2 because 2 is two units away from 0. To graph 2, we will place a closed circle at 2 on a number line.
Exercises 3 The absolute value of a number is the distance on a number line that number is from 0. We know that ∣-1∣=1 because -1 is one unit away from 0. To graph 1, we will place a closed circle at 1 on a number line.
Exercises 4 The absolute value of a number is the distance on a number line that number is from 0. For the given expression, we need to simplify before graphing. Since an absolute value represents a distance on the number line from 0, it will always be non-negative. 2+∣-2∣∣-2∣=22+2Add terms4 Now that the expression has been simplified, we can graph 4 as a closed circle on a number line.
Exercises 5 The absolute value of a number is the distance on a number line that number is from 0. For the given expression, we need to simplify before graphing. Since an absolute value represents a distance on the number line from 0, it will always be non-negative. 1−∣-4∣∣-4∣=41−4Subtract terms-3 Now that the expression has been simplified, we can graph -3 as a closed circle on a number line.
Exercises 6 The absolute value of a number is the distance on a number line that number is from 0. For the given expression, we need to simplify before graphing. Since an absolute value represents a distance on the number line from 0, it will always be non-negative. -5+∣3∣∣3∣=3-5+3Add terms-2 Now that the expression has been simplified, we can graph -2 as a closed circle on a number line.
Exercises 7 We can use a number line to help us determine which of the two given numbers is larger. This is specifically helpful when we have two negative numbers. Numbers that lie further to the left on a number line are smaller, while numbers that lie toward the right are bigger. To begin, we will graph 2 and 9 on a number line.Since 2 lies to the left of 9, we can conclude that 2 is less than 9. Symbolically, we can write that as 2<9
Exercises 8 We can use a number line to help us determine which of the two given numbers is larger. This is specifically helpful when we have two negative numbers. Numbers that lie further to the left on a number line are smaller, while numbers that lie toward the right are bigger. To begin, we will graph -6 and 5 on a number line.Since -6 lies to the left of 5, we can conclude that -6 is less than 5. Symbolically, we can write that as -6<5
Exercises 9 We can use a number line to help us determine which of the two given numbers is larger. This is especially helpful when we have two negative numbers. Numbers that lie further to the left on a number line are smaller, while numbers that lie toward the right are bigger. To begin, we will graph -12 and -4 on a number line.Since -12 lies to the left of -4, we can conclude that -12 is less than -4. Symbolically, we can write that as -12<-4
Exercises 10 We can use a number line to help us determine which of the two given numbers is larger. This is especially helpful when we have two negative numbers. Numbers that lie further to the left on a number line are lesser, while numbers that lie toward the right are greater. To begin, we will graph -7 and -13 on a number line.Since -13 lies to the left of -7, we can conclude that -7 is greater than -13. -7>-13​
Exercises 11 To determine which of the given values is larger than the other, we will begin by determining the absolute values of the numbers. The absolute value of any number is the distance on a number line from that number to 0. Since the distance from 0 to both -8 and 8 is 8 We know that ∣-8∣=8and∣8∣=8.​ Thus, the values are equal. Symbolically we can write this as ∣-8∣=∣8∣.​
Exercises 12 To determine which of the given values is larger, we will begin by determining the absolute values of the second number. The absolute value of any number is the distance on a number line from 0 to that number. Since the distance from 0 to 18 is 18 we have that ∣-18∣=18.​ We can use a number line to determine which of the two given numbers is larger. Numbers that lie further out on the left side on a number line are smaller, while numbers that lie towards the right are bigger.Since -10 lies to the left of 18, we can conclude that -10 is less than ∣-18∣. Symbolically, we can write that as -10<∣-18∣​
Exercises 13 We know that numbers that lie toward the left on a number line are smaller, while numbers that lie toward the right are bigger. It is given that a lies to the left of b which means a<b. For our specific example, we can let a and b equal any values such that a lies to the left of b. We will arbitrarily choose a=2 and b=8. This gives the following number line.If a=2 and b=8 then we have -a=-2 and -b=-8. Let's add these numbers to the number line.Since -b lies to the left of -a, we have that -b<-a. This will hold true for all possible values of a and b. Also, this is why you must reverse the inequality sign when you either divide or multiply an inequality by -1.