If a<2b−1 , b<c+2 , and c<3a−4, then what is the least positive integer value for a?

Given the inequalities:

1. \( a < 2b - 1 \)
2. \( b < c + 2 \)
3. \( c < 3a - 4 \)

we need to find the smallest positive integer value for \( a \) that satisfies all these conditions.

Let's proceed by testing small integer values of \( a \) to see if we can find a feasible solution.

### Step-by-Step Solution

#### Step 1: Assume \( a = 1 \)
If \( a = 1 \), we substitute into the first inequality:
\[
1 < 2b - 1 \Rightarrow 2 < 2b \Rightarrow b > 1
\]
Thus, \( b \) must be at least 2 for \( a = 1 \).

Next, substitute \( b = 2 \) into the second inequality:
\[
2 < c + 2 \Rightarrow c > 0
\]
Thus, \( c \) must be at least 1.

Finally, substitute \( a = 1 \) and \( c = 1 \) into the third inequality:
\[
1 < 3(1) - 4 \Rightarrow 1 < -1
\]
This is not possible, so \( a = 1 \) does not satisfy the conditions.

#### Step 2: Assume \( a = 2 \)
If \( a = 2 \), we substitute into the first inequality:
\[
2 < 2b - 1 \Rightarrow 3 < 2b \Rightarrow b > 1.5
\]
Thus, \( b \) must be at least 2.

Next, substitute \( b = 2 \) into the second inequality:
\[
2 < c + 2 \Rightarrow c > 0
\]
Thus, \( c \) must be at least 1.

Finally, substitute \( a = 2 \) and \( c = 1 \) into the third inequality:
\[
1 < 3(2) - 4 \Rightarrow 1 < 2
\]
This is true, so \( a = 2 \) satisfies all conditions.

### Conclusion
The smallest positive integer value for \( a \) that satisfies all the inequalities is:
\[
\boxed{2}
\]

Would you like more details or have any questions?

---

**Further Exploration:**

1. How would the answer change if there were additional constraints on \( b \) or \( c \)?
2. What would be the effect on the solution if one of the inequalities were non-strict?
3. Can you explain how inequalities relate to systems of linear equations?
4. How does testing small values help solve inequality problems?
5. Could this problem be solved graphically?

**Tip:** When solving inequalities, break down each step carefully, as small errors can lead to incorrect conclusions.

If a < 2b - 1, b < c + 2, and c < 3a - 4, then what is the least positive integer value for a?

Learn how to solve a system of inequalities to find the smallest positive integer for a, satisfying a < 2b - 1, b < c + 2, and c < 3a - 4. This problem involves understanding algebraic inequalities and testing integer values, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation