Solve for all values of bb in simplest form.
30, equals, open vertical bar, 10, plus, 4, b, close vertical bar
30=
∣10+4b∣

We are given the equation:

\[
30 = |10 + 4b|
\]

To solve this, we need to break it down into two cases since absolute value equations are defined based on the two possible scenarios: positive and negative values inside the absolute value function.

### Case 1: Positive scenario
In this case, we assume that \(10 + 4b \geq 0\). Therefore, we can drop the absolute value and solve:

\[
30 = 10 + 4b
\]

Now, subtract 10 from both sides:

\[
30 - 10 = 4b
\]

\[
20 = 4b
\]

Now, divide by 4 to solve for \(b\):

\[
b = \frac{20}{4} = 5
\]

### Case 2: Negative scenario
In this case, we assume that \(10 + 4b < 0\). Therefore, we take the negative of the expression inside the absolute value:

\[
30 = -(10 + 4b)
\]

Simplify the right-hand side:

\[
30 = -10 - 4b
\]

Now, add 10 to both sides:

\[
30 + 10 = -4b
\]

\[
40 = -4b
\]

Now, divide by -4 to solve for \(b\):

\[
b = \frac{40}{-4} = -10
\]

### Final Answer:
The two solutions are:

\[
b = 5 \quad \text{or} \quad b = -10
\]

Would you like more details on this process or have any questions?

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Here are 5 related questions you might find helpful:
1. How do you solve absolute value equations with inequalities?
2. What is the general process for solving absolute value equations?
3. How do you check the solutions of an absolute value equation?
4. Can you explain how absolute values affect linear equations?
5. How can we graph absolute value functions and interpret their solutions?

**Tip:** Always check both cases (positive and negative) when solving absolute value equations to avoid missing solutions!

Solve for all values of b in simplest form. 30 = |10 + 4b|

Learn how to solve the absolute value equation 30 = |10 + 4b| with a step-by-step guide. This solution explores both the positive and negative scenarios and provides solutions for b = 5 or b = -10. Perfect for students in Grades 7-9 learning about absolute value and linear equations.

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