MATH SOLVE

5 months ago

Q:
# 2. Another machine is used to make 10 mm long bolts. For this machine, the bolts must have an error of less than 0.1 mm to be accepted. This can be represented by |x – 10| < 0.1.a. Write this inequality as a compound inequality without absolute value bars.b. Solve the compound inequality.c. Graph the solutions on a number line.d. What do the solutions of the inequality mean for the bolt?3. Special precision bolts are made by another machine. These are for jobs that require more exact measurements. The bolts must be within 0.01 mm of 8 mm. This is represented by |x – 8| < 0.01. a. Write this inequality as a compound inequality without absolute value bars.b. Solve the inequality.c. Graph the solutions on a number line.d. What do the solutions of the inequality mean for the bolt?4. Suppose 1.5-inch long bolts are made by another machine and have to be within 0.125 inches of 1.5 inches.a. Write an absolute value inequality to represent this tolerance.b. Rewrite the absolute value inequality as a compound inequality without absolute value bars.c. Solve the compound inequality.d. What does the solution mean for the bolts?30 points!!! need help asape. Give an example of one acceptable length for a bolt.f. Give an example of one non-acceptable length for a bolt.

Accepted Solution

A:

Problem 2

Part A

Answer: -0.1 < x-10 < 0.1

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

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Part B

Answer: 9.9 < x < 10.1

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

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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"

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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

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Problem 3

Part A

Answer: -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

Answer: 7.99 <= x <= 8.01

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

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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"

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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

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Problem 4

Part A

Answer: |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

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Part B

Answer: -0.125 <= x-1.5 <= 0.125

Same rule as before

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Part C

Answer: 1.375 <= x <= 1.625

Add 1.5 to all sides of the answer from part B

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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

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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

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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

Part A

Answer: -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

Answer: 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

Answer: -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

Answer: 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

Answer: |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

Answer: -0.125 <= x-1.5 <= 0.125

Same rule as before

---------------

Part C

Answer: 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