In the inequalities, a and b are integers. What is the least integer value of a/bâ€â€¹? −5≤a≤5 1<b<5
-2
To find the least integer value of \( \frac{a}{b} \) given the constraints \(-5 \leq a \leq 5\) and \(1 < b < 5\), we need to consider the possible integer values of \(b\) and the corresponding values of \(a\). The minimum ratio occurs when \(a\) is at its lowest value and \(b\) is at its highest value.
For \( \frac{a}{b} \) to equal 2, \(a\) would need to be at least 2 times \(b\). The highest possible value of \(b\) is 4 (since \(1 < b < 5\)), giving \(a\) a minimum value of 8, which is not allowed since \(a\) must be between -5 and 5. Thus, this choice is not feasible.
If \( \frac{a}{b} = -1\), then \(a\) would need to equal \(-b\). The maximum \(b\) can be is 4, which would yield \(a = -4\). Although \(-4\) is within the bounds for \(a\), it is not the least value achievable; therefore, this option is incorrect.
Setting \( \frac{a}{b} = -2\) requires \(a = -2b\). For \(b = 4\), \(a\) would be \(-8\), which is out of bounds. However, for \(b = 3\), \(a\) would equal \(-6\), still invalid. But for \(b = 2\), \(a = -4\) is valid, leading to the least integer value of \(-2\) for \( \frac{a}{b} \), making this the correct choice.
If \( \frac{a}{b} = -4\), then \(a\) would need to be \(-4b\). The maximum permissible value for \(b\) is 4, which would give \(a = -16\), clearly outside the specified range for \(a\). Hence, this choice is not possible.
The problem requires finding the least integer value of \( \frac{a}{b} \) under the given constraints. After evaluating all options, it is evident that \(-2\) is the least achievable ratio when \(a = -4\) and \(b = 2\). Other choices either exceed the limits for \(a\) or do not yield a lesser value, confirming that \(-2\) is indeed the correct answer.
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