Question

Caleb bought some cat food for $12 and 4 toys for x dollars each. He spent less than $41 total. How much could each toy have cost?
Write an inequality to represent the situation. Use x to represent the cost of each toy.
Excellent!
You just wrote an inequality to represent the situation. Now solve the associated equation, 12 + 4x = 41, to find the boundary for the inequality. Show your work or solve on paper.
12 + 4x < 41
12+4x
=
41
−
−
=

x
=
Excellent!
You found that x = 7.25 is the solution to the associated equation and the boundary of the inequality. Does the boundary, x = 7.25, make the inequality true or false?
x= 7 x=7.25 (boundary) x= 8
12 + 4x < 41 ?
?

Answer 1

To determine whether the boundary \( x = 7.25 \) makes the inequality \( 12 + 4x < 41 \) true or false, we need to evaluate the expression for both \( x = 7 \) and \( x = 8 \) in addition to the boundary value \( x = 7.25 \).

First, let's restate the inequality:

\[
12 + 4x < 41
\]

Now, solving the associated equation \( 12 + 4x = 41 \):

1. Subtract 12 from both sides:
\[
4x = 41 - 12
\]
\[
4x = 29
\]

2. Divide by 4:
\[
x = \frac{29}{4} = 7.25
\]

Now, let's test the values:

1. **For \( x = 7 \)**:
\[
12 + 4(7) = 12 + 28 = 40
\]
Since \( 40 < 41 \), this is **true**.

2. **For \( x = 7.25 \)** (boundary):
\[
12 + 4(7.25) = 12 + 29 = 41
\]
Since \( 41 < 41 \) is **false**, this value does not satisfy the inequality.

3. **For \( x = 8 \)**:
\[
12 + 4(8) = 12 + 32 = 44
\]
Since \( 44 < 41 \) is **false**.

**Conclusion:**
- At \( x = 7 \), the inequality is true.
- At \( x = 7.25 \), the boundary point does not satisfy the inequality (is false).
- At \( x = 8 \), the inequality is false.

Thus, the cost of each toy, \( x \), must be **less than 7.25** for the inequality \( 12 + 4x < 41 \) to hold true.