Which values of x are solutions of the compound inequality 4<4x

Problem 2

Part A
-0.1 < x-10 < 0.1

I'm using the rule |x| < k leads to -k < x < k for some positive number k



Part B
9.9 < x < 10.1

To find this answer, you add 10 to all three sides of -0.1 < x-10 < 0.1



Part C

Draw out a number line. Plot open circles at 9.9 and 10.1 on the number line. Shade the region between the open circles. The open circles indicate "do not include this point as part of the solution set"



Part D

The solution 9.9 < x < 10.1 means that anything between 9.9 and 10.1 is accepted as a bolt length (even if the exact measurement isn't 10 mm on the dot). The length 9.9 is not allowed; neither is 10.1



Problem 3

Part A

-0.01 <= x-8 <= 0.01

Use the same rule from problem 2, part A

Note: the inequality given was |x-8| < 0.01 but I have a feeling your teacher meant to say |x-8| <= 0.01
The reason why I'm thinking is this is due to the keywords "must be within"

The notation <= means "less than or equal to"



Part B

7.99 <= x <= 8.01

Similar to problem 2, part B, but this time we add 8 to all three sides to isolate x



Part C

Draw out a number line. Plot closed circles at 7.99 and 8.01; then shade the region between the closed circles. A closed circle means "include this point as a solution"



Part D

The solutions mean that anything between 7.99 mm and 8.01 mm is allowed. The value 7.99 is allowed; so is 8.01



Problem 4

Part A

|x-1.5| <= 0.125

Explanation: The target T and error E is such that |x-T| <= E. We want the distance/difference from the measured x to the target T to be E or less



Part B

-0.125 <= x-1.5 <= 0.125

Same rule as before



Part C

1.375 <= x <= 1.625

Add 1.5 to all sides of the answer from part B



Part D

The solution from part C means any length between 1.375 inches and 1.625 inches is allowed. The value 1.375 is allowed; so is 1.625



Part E

An acceptable length is 1.5 inches as this is between 1.375 and 1.625
In other words, x = 1.5 will make 1.375 <= x <= 1.625 a true inequality (replace x with 1.5)
So 1.375 <= 1.5 <= 1.625 is true

Note: there is no one set answer as long as it's between 1.375 and 1.625



Part F

A non-acceptable length is x = 10 as this is too large.
The value 10 is not between 1.375 and 1.625
The value x = 10 will make 1.375 <= x <= 1.625 false
In other words, 1.375 <= 10 <= 1.625 is false

Note: there is no one set answer for this as you can pick anything larger than 1.625, or anything smaller than 1.375

Problem 2

Part A
-0.1 < x-10 < 0.1

I'm using the rule |x| < k leads to -k < x < k for some positive number k



Part B
9.9 < x < 10.1

To find this answer, you add 10 to all three sides of -0.1 < x-10 < 0.1



Part C

Draw out a number line. Plot open circles at 9.9 and 10.1 on the number line. Shade the region between the open circles. The open circles indicate "do not include this point as part of the solution set"



Part D

The solution 9.9 < x < 10.1 means that anything between 9.9 and 10.1 is accepted as a bolt length (even if the exact measurement isn't 10 mm on the dot). The length 9.9 is not allowed; neither is 10.1



Problem 3

Part A

-0.01 <= x-8 <= 0.01

Use the same rule from problem 2, part A

Note: the inequality given was |x-8| < 0.01 but I have a feeling your teacher meant to say |x-8| <= 0.01
The reason why I'm thinking is this is due to the keywords "must be within"

The notation <= means "less than or equal to"



Part B

7.99 <= x <= 8.01

Similar to problem 2, part B, but this time we add 8 to all three sides to isolate x



Part C

Draw out a number line. Plot closed circles at 7.99 and 8.01; then shade the region between the closed circles. A closed circle means "include this point as a solution"



Part D

The solutions mean that anything between 7.99 mm and 8.01 mm is allowed. The value 7.99 is allowed; so is 8.01



Problem 4

Part A

|x-1.5| <= 0.125

Explanation: The target T and error E is such that |x-T| <= E. We want the distance/difference from the measured x to the target T to be E or less



Part B

-0.125 <= x-1.5 <= 0.125

Same rule as before



Part C

1.375 <= x <= 1.625

Add 1.5 to all sides of the answer from part B



Part D

The solution from part C means any length between 1.375 inches and 1.625 inches is allowed. The value 1.375 is allowed; so is 1.625



Part E

An acceptable length is 1.5 inches as this is between 1.375 and 1.625
In other words, x = 1.5 will make 1.375 <= x <= 1.625 a true inequality (replace x with 1.5)
So 1.375 <= 1.5 <= 1.625 is true

Note: there is no one set answer as long as it's between 1.375 and 1.625



Part F

A non-acceptable length is x = 10 as this is too large.
The value 10 is not between 1.375 and 1.625
The value x = 10 will make 1.375 <= x <= 1.625 false
In other words, 1.375 <= 10 <= 1.625 is false

Note: there is no one set answer for this as you can pick anything larger than 1.625, or anything smaller than 1.375

1)  x < 4  the answer is    Option C: x E(-infinity,4)  (it is strict)
2)  2x +6 < 8, and  2x < 8 - 6 = 2, and x<2/2=1
the answer is
D. x E (-infinity, 1)
3)  X greater than or equal to 8 and  x <  - 4
X greater than  8  means x ≥ 8
so we have x ≥ 8 and   x <  - 4
so the answer is 
B. x E (-4,8]
 4) 
7 > x + 6 or x -2 greater than or equal to 3,
7 > x + 6 or x -2 ≥ 3

7 -6> x, and 1>x

x -2 ≥ 3   x≥ 3+2=5
finally
x<1 and x≥5   

the answer is
C.x E (1,5]
5)

|x+3| > 12, 
|x+3| = { -(x+3) if x+3<0 or  (x+3)  if x+3>0}

so, it is  -(x+3)>12 is equivalent to (x+3)< -12  or (x+3)>12
the answer is
B. x + 3 > 12 or x + 3 <-12

1)  x < 4  the answer is    Option C: x E(-infinity,4)  (it is strict)
2)  2x +6 < 8, and  2x < 8 - 6 = 2, and x<2/2=1
the answer is
D. x E (-infinity, 1)
3)  X greater than or equal to 8 and  x <  - 4
X greater than  8  means x ≥ 8
so we have x ≥ 8 and   x <  - 4
so the answer is 
B. x E (-4,8]
 4) 
7 > x + 6 or x -2 greater than or equal to 3,
7 > x + 6 or x -2 ≥ 3

7 -6> x, and 1>x

x -2 ≥ 3   x≥ 3+2=5
finally
x<1 and x≥5   

the answer is
C.x E (1,5]
5)

|x+3| > 12, 
|x+3| = { -(x+3) if x+3<0 or  (x+3)  if x+3>0}

so, it is  -(x+3)>12 is equivalent to (x+3)< -12  or (x+3)>12
the answer is
B. x + 3 > 12 or x + 3 <-12

Since the deviation from the average speed of the cyclist can only be up to 8 miles per hour, in the first equation the value of x is,
                            -x < 31 - s < 8
The value of x is -8. For the second one,
                               x > 31 - s < 8 
The value of x should be 8.