A manufacturing tolerance specification requires a component dimension $L$ (in centimeters) to satisfy $|100L - 450| = 15$. Convert the tolerance equation to millimeters by substituting $L = \frac{y}{10}$, where $y$ is the length in millimeters, and solve for all acceptable values of $y$ in millimeters.

A manufacturing tolerance specification requires a component dimension $L$ (in centimeters) to satisfy $|100L - 450| = 15$ . Convert the tolerance equation to millimeters by substituting $L = \frac{y}{10}$ , where $y$ is the length in millimeters, and solve for all acceptable values of $y$ in millimeters.

Final answer

$y = 43.5\text{ mm}$ or $y = 46.5\text{ mm}$

Steps

Substitute the conversion relation into the equation

The original equation in centimeters is: $|100L - 450| = 15$

Substitute $L = \frac{y}{10}$ into the equation to express the tolerance in terms of the length $y$ in millimeters: $|100\left(\frac{y}{10}\right) - 450| = 15$

Simplify the expression inside the absolute value bars: $|10y - 450| = 15$

Set up the two linear cases

By definition, an equation of the form $|A| = c$ (where $c > 0$ ) splits into two separate linear cases: $A = c$ or $A = -c$ .

Case 1 (Upper limit): $10y - 450 = 15$ $10y = 450 + 15$ $10y = 465$ $y = \frac{465}{10} = 46.5$
Case 2 (Lower limit): $10y - 450 = -15$ $10y = 450 - 15$ $10y = 435$ $y = \frac{435}{10} = 43.5$

Alternative geometric interpretation (verification)

Factoring $10$ out of the absolute value expression gives: $10|y - 45| = 15 \implies |y - 45| = 1.5$

This means that the acceptable length $y$ is centered at $45\text{ mm}$ with a tolerance (distance) of at most $1.5\text{ mm}$ : $y = 45 \pm 1.5 \implies y = 43.5\text{ mm}\quad \text{or}\quad y = 46.5\text{ mm}$

Common Pitfall: Forgetting the negative case ( $10y - 450 = -15$ ) will result in finding only the upper tolerance limit and missing the lower limit. Always account for both directions from the center point when removing absolute value bars.

Thus, the acceptable values are $y = 43.5\text{ mm}$ or $y = 46.5\text{ mm}$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student solves only the positive case of the absolute value equation, omitting the negative case.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
|100(y/10) - 450| = 15 $\implies$ |10y - 450| = 15
Step 2
10y - 450 = 15
Correct
10y - 450 = 15 $\text{or}$ 10y - 450 = -15
Removing absolute value bars from $|A| = c$ (with $c > 0$ ) requires considering both branches: $A = c$ and $A = -c$ . You only wrote the positive case and omitted the negative case.
Step 3
10y = 465 $\implies$ y = 46.5 $\text{ mm}$
Correct
y = 46.5 $\text{ mm}$ $\text{or}$ y = 43.5 $\text{ mm}$
Your algebra for the single branch was solved accurately, but because the negative branch was omitted, your final solution set is missing $y = 43.5\text{ mm}$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly substituted $L = \frac{y}{10}$ into the tolerance equation.

You simplified the linear term inside the absolute value bars accurately to $10y$ .

You solved the linear equation $10y - 450 = 15$ with correct algebraic steps.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted negative case in absolute value equation

You dropped the absolute value bars directly without splitting the equation into two separate cases.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating an absolute value equality $|A| = c$ as simply $A = c$ , forgetting that distance from a center can be in either the positive or negative direction.

REASONING PATH

Your ACEpath teacher’s suggested approach

Two-Branch Definition: For any positive real number $c$ , the equation $|A| = c$ is equivalent to $A = c$ or $A = -c$ .
Whenever you remove absolute value bars from $|A| = c$ , immediately write two separate linear equations connected by 'or'.
Physical / Geometric Meaning: In manufacturing tolerances, $|y - \text{center}| = \text{tolerance}$ defines both an upper limit (center + tolerance) and a lower limit (center - tolerance).
Always verify that your final answer provides both bounds of the tolerance interval.

CORRECTED PATH

ACEpath Teacher's Demonstration

Set up Case 1: $10y - 450 = 15 \implies 10y = 465 \implies y = 46.5\text{ mm}$ .
Set up Case 2: $10y - 450 = -15 \implies 10y = 435 \implies y = 43.5\text{ mm}$ .

Final answer

$y = 43.5\text{ mm}$ or $y = 46.5\text{ mm}$

NEXT STEPS

Final Takeaways from Your Teacher

Review how absolute value represents distance on a number line in both directions from a center point.
Practice splitting equations of the form $|ax + b| = c$ into $ax + b = c$ and $ax + b = -c$ before doing any arithmetic.
Complete the three targeted practice exercises below to build fluency with dual-case solutions.

OverviewACEpath Teacher's Diagnosis

02 The student incorrectly multiplies by 10 instead of dividing by 10 when substituting for L.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
|100(10y) - 450| = 15 $\implies$ |1000y - 450| = 15
Correct
$|100\left(\frac{y}{10}\right) - 450| = 15 \implies |10y - 450| = 15$
You substituted $10y$ in place of $L$ rather than the given relation $L = \frac{y}{10}$ . Because $1\text{ cm} = 10\text{ mm}$ , a length of $y\text{ mm}$ equals $\frac{y}{10}\text{ cm}$ .
Step 2
1000y - 450 = 15 $\text{ or }$ 1000y - 450 = -15 $\implies$ 1000y = 465 $\text{ or }$ 1000y = 435
Correct
10y - 450 = 15 $\text{ or }$ 10y - 450 = -15 $\implies$ 10y = 465 $\text{ or }$ 10y = 435
Your method for setting up and isolating both linear branches was mathematically sound, but the numbers were affected by the incorrect coefficient from Step 1.
Step 3
y = 0.465 $\text{ mm or }$ y = 0.435 $\text{ mm}$
Correct
y = 46.5 $\text{ mm or }$ y = 43.5 $\text{ mm}$
Because of the extra factor of 100 introduced during substitution, your final values are off by a factor of 100.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly split the absolute value equation $|A| = c$ into its positive and negative linear branches.

You accurately solved both linear branches for the variable.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incorrect Variable Substitution / Unit Conversion Inversion

You multiplied by 10 instead of dividing by 10 when replacing $L$ with $y$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the unit relation ( $1\text{ cm} = 10\text{ mm}$ ) with the algebraic substitution formula ( $L = \frac{y}{10}$ ), leading to multiplying by the conversion factor rather than substituting the exact fractional expression given.

REASONING PATH

Your ACEpath teacher’s suggested approach

Literal Substitution: When substituting an expression for a variable, enclose the exact given expression in parentheses.
Write out $L = \left(\frac{y}{10}\right)$ and replace $L$ directly without inverting or multiplying in your head.
Sanity Check on Units: A measurement expressed in smaller units (millimeters) must have a larger numerical value than in larger units (centimeters).
Since $L \approx 4.5\text{ cm}$ , the millimeter value $y$ must be around $45\text{ mm}$ , not $0.45\text{ mm}$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Substitute $L = \frac{y}{10}$ into $|100L - 450| = 15$ : $|100\left(\frac{y}{10}\right) - 450| = 15$ .
Simplify the coefficient inside: $|10y - 450| = 15$ .
Split into two equations: $10y - 450 = 15$ or $10y - 450 = -15$ .
Solve: $10y = 465 \implies y = 46.5$ , and $10y = 435 \implies y = 43.5$ .

Final answer

y = 43.5 $\text{ mm or }$ y = 46.5 $\text{ mm}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice substituting algebraic fractions directly using parentheses before simplifying coefficients.
Perform a physical magnitude check: verify that millimeters are numerically 10 times larger than centimeters.

OverviewACEpath Teacher's Diagnosis

03 The student misapplies the absolute value rule by writing 10y + 450 = 15 instead of setting the entire expression to -15.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$|100(y/10) - 450| = 15 \implies |10y - 450| = 15$
Step 2
$10y - 450 = 15 \text{ or } 10y + 450 = 15$
Correct
$10y - 450 = 15 \text{ or } 10y - 450 = -15$
To solve $|A| = c$ , you must set $A = c$ or $A = -c$ . You changed the minus sign inside the binomial to a plus sign ( $10y + 450 = 15$ ) instead of negating the constant on the right-hand side.
Step 3
$10y = 465 \text{ or } 10y = -435$
Correct
$10y = 465 \text{ or } 10y = 435$
Your algebraic solving steps followed your equations correctly, but the second equation carried the error from step 2.
Step 4
$y = 46.5\text{ mm} \text{ or } y = -43.5\text{ mm}$
Correct
$y = 46.5\text{ mm} \text{ or } y = 43.5\text{ mm}$
Because of the setup error in the second branch, your second solution resulted in $-43.5\text{ mm}$ instead of $43.5\text{ mm}$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly substituted the conversion ratio $L = \frac{y}{10}$ into the tolerance equation.

You accurately simplified the expression inside the absolute value bars to $10y - 450$ .

You solved the linear branch $10y - 450 = 15$ accurately to find $y = 46.5\text{ mm}$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Absolute value case splitting error

You changed $-450$ to $+450$ inside the expression instead of negating the other side of the equation.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the two branches of an absolute value equation by altering an internal sign rather than applying the definition $|A| = c \iff A = c \text{ or } A = -c$ (or $-(A) = c$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Absolute Value Definition: For any real expression $A$ and constant $c > 0$ , $|A| = c$ means $A = c$ or $A = -c$ .
Keep the entire expression inside the bars completely unchanged and set it equal to both $+c$ and $-c$ .
Physical Reality Check: Component lengths and dimensions in manufacturing cannot be negative.
When you obtain a negative value like $y = -43.5\text{ mm}$ for a physical dimension, pause and verify your case setup.

CORRECTED PATH

ACEpath Teacher's Demonstration

Set up Case 1: $10y - 450 = 15$
Solve Case 1: $10y = 465 \implies y = 46.5\text{ mm}$
Set up Case 2: $10y - 450 = -15$
Solve Case 2: $10y = 450 - 15 = 435 \implies y = 43.5\text{ mm}$

Final answer

$y = 43.5\text{ mm}$ or $y = 46.5\text{ mm}$

NEXT STEPS

Final Takeaways from Your Teacher

Review the formal definition of absolute value as distance from zero on a number line.
Practice rewriting $|A| = c$ as $A = c \text{ or } A = -c$ without modifying the terms inside $A$ .
Perform a sanity check on real-world dimension values to catch impossible negative lengths.

Lock it in

Question

What two linear equations must you write to solve $|A| = c$ when $c > 0$ ?

Answer

$A = c$ or $A = -c$ . The absolute value represents distance, so the quantity inside can be either $c$ units to the right or $c$ units to the left of zero.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student solves only the positive case of the absolute value equation, omitting the negative case.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student incorrectly multiplies by 10 instead of dividing by 10 when substituting for L.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student misapplies the absolute value rule by writing 10y + 450 = 15 instead of setting the entire expression to -15.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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