Chapter 19: Problem 5
For all values of
Short Answer
Expert verified
The correct order is , so the answer is option d.
Step by step solution
01
- Compute R
Calculate the expression for using the given operation definition. Substitute , , and .
02
- Compute S
Calculate the expression for using the given operation definition. Substitute , , and .
03
- Compute T
Calculate the expression for using the given operation definition. Substitute , , and .
04
- Analyze signs for comparison
Given that , evaluate the expressions: : since , . : since , . : since , .
05
- Compare magnitudes
Since both and are negative and is positive, and are less than . To compare and : is more negative than because the coefficient 27 is larger.Therefore, .
06
- Arrange values
Arrange the values in order from least to greatest: .
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
custom operations
Custom operations are functions or operations defined outside of standard arithmetic rules. In this exercise, the custom operation is defined as:
) 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: , , and into the custom operation formula, we get:
x \diamond y \diamond z = x^{2}(y-1)(z+2)This operation combines three variables (
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.
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 :
:
:
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.
After substituting the variables into the custom operation formula, you need to simplify these expressions further. Example, for the term
R = 1^{2}(a-1)(3+2) = (a-1) \times 5 = 5(a-1)Similarly for
S = 3^{2}(a-1)(1+2) = 9(a-1) \times 3 = 27(a-1)And also for
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 . This information is essential for analyzing and comparing the computed values of and .
Given the values of are negative:
.
Understanding inequalities and how to apply them to algebraic expressions is important for solving similar GMAT problems.
Given the values of
because . because . since squared terms are always non-negative, and the multiplication by a positive constant keeps positive.
- Firstly, since
, is automatically greater than both and . - Next, compare
and . Because and , and we know is a negative factor, multiplying it by a larger coefficient results in a more negative value: .
Understanding inequalities and how to apply them to algebraic expressions is important for solving similar GMAT problems.