An absolute value equation of the form $|a x + b| = c$ has the exact solution set $\{-4, 10\}$, where $a = 1$ and $c > 0$. Determine the values of the constants $b$ and $c$.

An absolute value equation of the form $|a x + b| = c$ has the exact solution set $\{-4, 10\}$ , where $a = 1$ and $c > 0$ . Determine the values of the constants $b$ and $c$ .

Final answer

$b = -3$ , $c = 7$

Steps

To find the constants $b$ and $c$ in the equation $|ax + b| = c$ , substitute $a = 1$ :

$|x + b| = c$

Method 1: Geometric Interpretation (Midpoint and Distance)

Recall that the expression $|x - k|$ represents the distance between $x$ and a fixed center point $k$ on the number line. Rewriting the equation in this form:

$|x - (-b)| = c$

This means that the solutions are points on a number line located at a distance of $c$ units away from the center $-b$ .

Find the center ( $-b$ ):
The center is the midpoint of the two solutions $-4$ and $10$ : $-b = \frac{-4 + 10}{2} = \frac{6}{2} = 3$ Multiplying by $-1$ gives: $b = -3$
Find the distance ( $c$ ):
The constant $c$ is half the total distance between the two solutions (or the distance from the center to either solution): $c = \frac{10 - (-4)}{2} = \frac{14}{2} = 7$

Method 2: Algebraic System

By definition of the absolute value, the equation $|x + b| = c$ splits into two linear equations: $x + b = c \quad \text{or} \quad x + b = -c$

Solving each for $x$ : $x = c - b \quad \text{and} \quad x = -c - b$

Since $c > 0$ , the larger root is $c - b$ and the smaller root is $-c - b$ :

c - b &= 10 \\ -c - b &= -4 \end{aligned}$$

Add the two equations: $(c - b) + (-c - b) = 10 + (-4)$ $-2b = 6 \implies b = -3$
Subtract the second equation from the first: $(c - b) - (-c - b) = 10 - (-4)$ $2c = 14 \implies c = 7$

Both methods confirm that $b = -3$ and $c = 7$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student confuses the midpoint of the solutions with b instead of -b, taking the center of the interval directly as b = 3.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
|x + b| = c ⟹ $\text{Midpoint: }$ $\frac{-4 + 10}{2}$ = 3 ⟹ b = 3
Correct
|x + b| = |x - (-b)| = c ⟹ $\text{Midpoint }$ = -b = $\frac{-4 + 10}{2}$ = 3 ⟹ b = -3
The expression $|x + b|$ represents the distance between $x$ and the center point $-b$ , because it can be written as $|x - (-b)|$ . While you correctly calculated the midpoint of the solutions as $3$ , that center equals $-b$ , which gives $b = -3$ rather than $b = 3$ .
Step 2
$\text{Distance: }$ $\frac{10 - (-4)}{2}$ = $\frac{14}{2}$ = 7 ⟹ c = 7

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly recognized that the solutions to $|x + b| = c$ are symmetric around a midpoint.

You calculated the midpoint $\frac{-4 + 10}{2} = 3$ accurately.

You correctly found the distance from the midpoint to each solution, concluding $c = 7$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign error in geometric center definition

You equated $b$ directly to the midpoint $3$ instead of setting the center $-b = 3$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the form $|x - h| = c$ with $|x + b| = c$ . On a number line, distance between $x$ and a center $h$ is $|x - h|$ . Therefore, $|x + b| = |x - (-b)|$ , meaning the center is located at $-b$ , not $b$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Center-Radius Form Check: The absolute value expression $|x - h| = c$ has center $h$ and radius $c$ . For $|x + b| = c$ , the center is $h = -b$ .
Always rewrite $|x + b|$ as $|x - (-b)|$ before matching the midpoint to the constant.
Endpoint Verification Check: Testing either given root into $|x + b| = c$ must yield a true statement.
Substitute $x = 10$ : $|10 + (-3)| = |7| = 7$ , which matches $c = 7$ . Testing $b = 3$ gives $|10 + 3| = 13 \neq 7$ , immediately catching the sign error.

CORRECTED PATH

ACEpath Teacher's Demonstration

Rewrite $|x + b| = c$ in standard distance form: $|x - (-b)| = c$ .
Set the center equal to the calculated midpoint: $-b = 3$ .
Solve for $b$ by multiplying both sides by $-1$ : $b = -3$ .

Final answer

b = -3, c = 7

NEXT STEPS

Final Takeaways from Your Teacher

Review the geometric definition of absolute value as the distance between two points, $|x_1 - x_2|$ .
Practice converting forms like $|x + 5| = d$ into $|x - (-5)| = d$ to avoid sign confusion with the center.
Plug your final constants back into the original absolute value equation with both roots to verify your signs.

OverviewACEpath Teacher's Diagnosis

02 The student confuses the distance parameter c with the total distance between the two solutions instead of half the distance.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
c = 10 - (-4) = 14
Correct
c = $\frac{10 - (-4)}{2}$ = 7
The quantity $10 - (-4) = 14$ is the full span between the two solutions. Because the absolute value equation $|x - (-b)| = c$ measures distance from the midpoint to each solution, $c$ must be half of this total distance.
Step 2
-b = (-4 + 10)/2 = 3 $\implies$ b = -3
Step 3
b = -3,; c = 14
Correct
b = -3,; c = 7
Your value for $b$ is correct, but your final answer carries forward the unhalved value of $c$ from Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You recognized that rewritten as $|x - (-b)| = c$ , the center point is the midpoint of the two solution values.

You correctly calculated the average of $-4$ and $10$ to find $-b = 3$ , leading to $b = -3$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distance vs. Span Confusion

You calculated the total distance across both roots ( $14$ ) rather than dividing by $2$ to get the radius distance from the center.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the absolute value radius parameter $c$ in $|x - k| = c$ with the total diameter/span between the two solutions, forgetting that $c$ represents the distance from the midpoint $k$ to each endpoint.

REASONING PATH

Your ACEpath teacher’s suggested approach

Geometric meaning of |x - k| = c: In the equation $|x - k| = c$ , $k$ is the center (midpoint) and $c$ is the distance from $k$ to each solution.
Always compute $c = \frac{\text{larger root} - \text{smaller root}}{2}$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Find the total distance between the roots: $10 - (-4) = 14$ .
Divide the total distance by $2$ to find the distance from the center: $c = \frac{14}{2} = 7$ .

Final answer

b = -3,; c = 7

NEXT STEPS

Final Takeaways from Your Teacher

Review the number line representation of $|x - k| = c$ where $c$ represents the 'radius' (half-span).
Practice checking your results by substituting both roots back into $|x + b| = c$ to verify that each yields $|-4 - 3| = 7$ and $|10 - 3| = 7$ .

OverviewACEpath Teacher's Diagnosis

03 The student sets x + b = c directly for both roots without taking the negative case for the smaller root, solving 10 + b = c and -4 + b = c.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
10 + b = c $\text{and}$ -4 + b = -c
Step 2
$\text{Subtracting gives }$ 14 = 2c, $\text{ so }$ c = 7
Step 3
10 + b = -7
Correct
10 + b = 7 $\implies$ b = -3
Since your first equation was $10 + b = c$ and you found $c = 7$ , substituting $c$ requires setting $10 + b = 7$ , which gives $b = -3$ . You wrote $-7$ instead of $7$ .
Step 4
b = -17, c = 7
Correct
b = -3, c = 7
Your final value for $b$ is affected by the sign error in the back-substitution step.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly recognized that $|x + b| = c$ yields two linear equations, matching the larger root $10$ with $c$ and the smaller root $-4$ with $-c$ .

You correctly subtracted the two equations to solve for $c = 7$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Back-substitution sign slip

When substituting $c = 7$ back into your equation $10 + b = c$ , you wrote $-7$ instead of $7$ on the right-hand side.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Losing track of the paired signs when back-substituting into a system of equations, or confusing the equation paired with $+c$ with the equation paired with $-c$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Equation Matching Check: When substituting a solved variable back into a system, double-check which equation has $+c$ and which has $-c$ .
Point to the original line $10 + b = c$ and verify that $c$ is replaced directly by $+7$ , not $-7$ .
Geometric Midpoint Sanity Check: In $|x + b| = c$ , the center is $-b$ , which must equal the midpoint of the two roots $\frac{x_1 + x_2}{2}$ .
Compute $\frac{10 + (-4)}{2} = 3$ , so $-b = 3 \implies b = -3$ to catch algebraic slips immediately.

CORRECTED PATH

ACEpath Teacher's Demonstration

Replace $c$ with $7$ in $10 + b = c$ : $10 + b = 7$
Subtract $10$ from both sides: $b = 7 - 10 = -3$
Optionally check with the second equation: $-4 + (-3) = -7 = -c$ This holds true since $-c = -7$ .

Final answer

b = -3, c = 7

NEXT STEPS

Final Takeaways from Your Teacher

Use the midpoint formula $\text{midpoint} = -b$ as a quick 5-second check whenever you solve for absolute value parameters.
When back-substituting, copy the exact equation label and formula before inserting numbers to prevent sign flips.

Lock it in

Question

What does the equation $|x - h| = c$ represent geometrically on a number line when $c > 0$ ?

Answer

It represents all points $x$ whose distance from the center $h$ is exactly $c$ , meaning the solutions are $h - c$ and $h + c$ .

Resources

Answer and steps

Final answer

Steps

Method 1: Geometric Interpretation (Midpoint and Distance)

Method 2: Algebraic System

Typical errors

01 The student confuses the midpoint of the solutions with b instead of -b, taking the center of the interval directly as b = 3.

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 confuses the distance parameter c with the total distance between the two solutions instead of half the distance.

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 sets x + b = c directly for both roots without taking the negative case for the smaller root, solving 10 + b = c and -4 + b = c.

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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