In the 𝑥𝑦-plane, if zero, zero is a solution to the system of inequalities, 𝑦 is greater than 𝑥 plus 𝑎 and 𝑦 is less than negative 𝑥 plus 𝑏, which of the following relationships between 𝑎 and 𝑏 must be true? Is it A) 𝑎 is greater than 𝑏, B) 𝑏 is greater than 𝑎, C) the absolute value of 𝑎 is greater than the absolute value of 𝑏, or D) 𝑎 is equal to negative 𝑏?
The first thing that we’re told is that zero, zero is a solution to the inequalities. This means that we can substitute 𝑥 equals zero and 𝑦 equals zero into the two inequalities. Substituting them into the first inequality gives us zero is greater than zero plus 𝑎. This simplifies to zero is greater than 𝑎. Which would more commonly be written as 𝑎 is less than zero. This means that 𝑎 can be any negative value.
Substituting zero, zero into the second inequality gives us zero is less than negative zero plus 𝑏. This simplifies to zero is less than 𝑏. Once again, we can rewrite this as 𝑏 is greater than zero. This means that 𝑏 can be any positive value. As zero, zero is a solution to the system of inequalities, 𝑎 must be negative and 𝑏 must be positive. If we consider a number line, then 𝑎 must be to the left of zero and 𝑏 must be to the right. We can write this as one inequality. Zero is greater than 𝑎 but less than 𝑏. This, in turn, means that 𝑎 must be less than 𝑏.
Looking at our four options, we can immediately rule out option A, as this stated that 𝑎 was greater than 𝑏. If 𝑎 is less than 𝑏, it cannot be greater than 𝑏. Option B must be correct. If 𝑎 is less than 𝑏, then 𝑏 must be greater than 𝑎. As 𝑏 is a positive value and 𝑎 is a negative value, then 𝑏 is greater than 𝑎.
Options C and D could possibly be true. However, the question asked, which relationship must be true? We can look at one example of option C and D where they work and one where they don’t work. As 𝑎 must be negative and 𝑏 must be positive, let’s consider the example when 𝑎 is negative four and 𝑏 is two. Option C states that the absolute value of 𝑎 is greater than the absolute value of 𝑏. In this case, the absolute value of negative four is greater than the absolute value of two. As the absolute value of negative four is four and the absolute value of two is two, this statement is correct. We have found one example where option C is correct.
As 𝑎 is negative and 𝑏 is positive, another solution is when 𝑎 is equal to negative three and 𝑏 is equal to three. This time, we have the absolute value of negative three is greater than the absolute value of three. This states that three is greater than three, which is not correct. Whilst there are some values of 𝑎 and 𝑏 where the modulus of 𝑎 is greater than the modulus of 𝑏, this is not true for all values.
Using the same two sets of values, we can see that option D, 𝑎 is equal to negative 𝑏, once again works in some cases. When 𝑎 is equal to negative three and 𝑏 is equal to three, this is true. However, when 𝑎 is equal to negative four and 𝑏 is equal to two, the statement is not true. This means that option D is not true for all values of 𝑎 and 𝑏. The only one of the four options that is true for all values is 𝑏 is greater than 𝑎.