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Chapter 1: Problem 4

The ordered pair \((3,-1)\) satisfies which of the following inequalities? I. \(x+3 y>0\) II. \(2 x+3 y>2\) III. \(x+y<0\) A) I only B) II only C) I and III only D) II and III only

Short Answer

Expert verified
The ordered pair \((3,-1)\) satisfies inequality II only. The correct answer is B) II only.

Step by step solution

01

Substitute the ordered pair \((3,-1)\) into each inequality.

Substitute \(x=3\) and \(y=-1\) into each inequality and evaluate the equations: 1. \(x+3y > 0\) 2. \(2x +3y > 2\) 3. \(x+y < 0\)
02

Evaluate inequality I.

For inequality I, substitute \(x=3\) and \(y=-1\): \(3 + 3(-1) > 0\) → \(3 - 3 > 0\) → \(0 > 0\) Inequality I is false, as \(0\) is not greater than \(0\).
03

Evaluate inequality II.

For inequality II, substitute \(x=3\) and \(y=-1\): \(2(3) + 3(-1) > 2\) → \(6 - 3 > 2\) → \(3 > 2\) Inequality II is true, as \(3\) is greater than \(2\).
04

Evaluate inequality III.

For inequality III, substitute \(x=3\) and \(y=-1\): \(3 + (-1) < 0\) → \(3 - 1 < 0\) → \(2 < 0\) Inequality III is false, as \(2\) is not less than \(0\).
05

Identify the satisfied inequalities.

As we have evaluated all three inequalities, we find that only inequality II is true. Thus, the ordered pair \((3,-1)\) satisfies inequality II only.
06

Find the correct answer.

Since the ordered pair \((3,-1)\) satisfies only inequality II, the correct answer is B) II only.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
In mathematics, an ordered pair consists of two elements with a particular order—typically represented as \( (x, y) \). These pairs are fundamental in coordinate geometry, as they pinpoint exact locations on a coordinate plane.

For example, in the given problem, the ordered pair \( (3, -1) \) is mentioned. The first number '3' represents the x-coordinate, and the second number '-1' represents the y-coordinate. To determine which inequalities the ordered pair satisfies, we substitute these coordinates into the given expressions and perform inequality evaluation.

Understanding how to work with ordered pairs is vital for mathematical topics ranging from algebraic equations to geometric interpretations on graphs. A strong grasp of this concept can greatly aid in visualizing and solving problems involving plotting points and interpreting graphs.
Inequality Evaluation
The process of inequality evaluationng involves substituting values into an inequality and determining whether the statement is true or false.

Consider the ordered pair from the exercise, \( (3, -1) \), which we substitute into each of the inequalities as the values of \(x\) and \(y\), respectively. To correctly evaluate inequalities,
  • Maintain the order of operations (PEMDAS)
  • Remember that when multiplying or dividing by a negative number, reverse the inequality sign
  • Keep the variables on one side and the constants on the other, if necessary
After substituting, simplify the expressions to find a true or false statement relative to the inequality presented.

Practicing inequality evaluation is crucial for testing numerical relationships and is a significant element in subjects like algebra and calculus, as well as in various standardized tests.
SAT Math Problems
Inequality questions are a common feature in SAT math problems, as they assess a student's ability to understand and manipulate algebraic expressions. These problems require students to apply mathematical concepts and perform calculations to find the solutions.

The sample question from the SAT provided deals with evaluating inequalities, which is a skill that can impact a student's score significantly. For success on these types of questions:
  • Read each inequality carefully
  • Substitute given values precisely
  • Determine the truth value of the inequality
  • Eliminate incorrect answer choices methodically
It is essential to practice with sample SAT questions to become familiar with the types of problems and the format of the test. By developing strategies for evaluating inequalities and other algebraic concepts, students can improve their performance and confidence on the SAT.

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