Question

For all values of $x,y,$ and $z,x\diamond y\diamond z=x^2(y-1)(z+2)$. If $a<0,$ which of the following shows $R,S,$ and $T$ arranged in order from least to greatest? $R:1\diamond a\diamond 3$ $S:3\diamond a\diamond 1$ $T:a\diamond 3\diamond 1$ a. $R,S,T$ b. $T,S,R$ c. $R,T,S$ d. $T,R,S$ e. $S,R,T$

Short answer

The correct order is $S,R,T$, so the answer is option d.

Step-by-step solution

- Compute R

Calculate the expression for $R=1\diamond a\diamond 3$ using the given operation definition. Substitute $x=1$, $y=a$, and $z=3$.$$R=1^2(a-1)(3+2)=(a-1)\times 5=5(a-1)$$

- Compute S

Calculate the expression for $S=3\diamond a\diamond 1$ using the given operation definition. Substitute $x=3$, $y=a$, and $z=1$.$$S=3^2(a-1)(1+2)=9(a-1)\times 3=27(a-1)$$

- Compute T

Calculate the expression for $T=a\diamond 3\diamond 1$ using the given operation definition. Substitute $x=a$, $y=3$, and $z=1$.$$T=a^2(3-1)(1+2)=a^2\times 2\times 3=6a^2$$

- Analyze signs for comparison

Given that $a<0$, evaluate the expressions:$R=5(a-1)$: since $a-1<0$, $R<0$.$S=27(a-1)$: since $a-1<0$, $S<0$.$T=6a^2$: since $a^2>0$, $T>0$.

- Compare magnitudes

Since both $R$ and $S$ are negative and $T$ is positive, $R$ and $S$ are less than $T$. To compare $R$ and $S$:$27(a-1)$ is more negative than $5(a-1)$ because the coefficient 27 is larger.Therefore, $S<R$.

- Arrange values

Arrange the values in order from least to greatest:$S<R<T$.

Key concepts

custom operations

Custom operations are functions or operations defined outside of standard arithmetic rules. In this exercise, the custom operation is defined as:

         x \diamond y \diamond z = x^{2}(y-1)(z+2) 

This operation combines three variables ($x,y,z$) through specific rules. Custom operations often appear in standardized tests like the GMAT to challenge your ability to adapt to new mathematical frameworks.
The key steps to solving problems with custom operations are:

• Understand and memorize the operation's definition.
• Substitute the given values into the operation accurately.
• Perform algebraic manipulations as indicated by the operation's formula.

For instance, when we substitute $x=1$, $y=a$, and $z=3$ into the custom operation formula, we get:

         R = 1 \diamond a \diamond 3 = 1^{2}(a-1)(3+2) = (a-1) \times 5 = 5(a-1) 

This substitution step follows directly from the custom rules defined for this problem.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. In this problem, algebraic manipulation is crucial for simplifying and comparing expressions.
After substituting the variables into the custom operation formula, you need to simplify these expressions further. Example, for the term $R$:

         R = 1^{2}(a-1)(3+2) = (a-1) \times 5 = 5(a-1) 

Similarly for $S$:

         S = 3^{2}(a-1)(1+2) = 9(a-1) \times 3 = 27(a-1) 

And also for $T$:

         T = a^{2}(3-1)(1+2) = a^{2} \times 2 \times 3 = 6a^{2} 

Understanding how to manipulate these expressions efficiently is a key skill in problem-solving. Notice how each term is simplified step-by-step by following the order of operations: first exponents, then multiplication inside the parentheses, and finally the overall multiplication.

inequalities

Inequalities involve comparing two expressions to determine which is greater or lesser. For this exercise, it is given that $a<0$. This information is essential for analyzing and comparing the computed values of $R,S,$ and $T$.

Given the values of $a$ are negative:

• $R=5(a-1)\to R<0$ because $a-1<0$.
• $S=27(a-1)\to S<0$ because $a-1<0$.
• $T=6a^2\to T>0$ since squared terms are always non-negative, and the multiplication by a positive constant keeps $T$ positive.

After identifying the signs of these expressions, the next step is comparing their magnitudes.

• Firstly, since $T>0$, $T$ is automatically greater than both $R$ and $S$.
• Next, compare $R$ and $S$. Because $R=5(a-1)$ and $S=27(a-1)$, and we know $(a-1)$ is a negative factor, multiplying it by a larger coefficient results in a more negative value: $S<R$.

So, arranging them from least to greatest:
$S<R<T$.
Understanding inequalities and how to apply them to algebraic expressions is important for solving similar GMAT problems.