/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Determine whether the ordered pa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 8

Determine whether the ordered pairs given are solutions. $$3 x-y>5 ;(0,0),(4,-1),(-1,-5),(1,-2)$$

Short Answer

Expert verified
Only (4,-1) is a solution.

Step by step solution

01

Understand the Inequality

We are given the inequality \(3x - y > 5\). We need to check whether each ordered pair satisfies this inequality.
02

Substitute (0,0) into the Inequality

For the pair (0,0), substitute \(x = 0\) and \(y = 0\) into the inequality \(3x - y > 5\). This gives \(3(0) - 0 > 5\), which simplifies to \(0 > 5\). This is false, so (0,0) is not a solution.
03

Substitute (4,-1) into the Inequality

For the pair (4,-1), substitute \(x = 4\) and \(y = -1\) into the inequality \(3x - y > 5\). This gives \(3(4) - (-1) > 5\), which simplifies to \(12 + 1 > 5\) or \(13 > 5\). This is true, so (4,-1) is a solution.
04

Substitute (-1,-5) into the Inequality

For the pair (-1,-5), substitute \(x = -1\) and \(y = -5\) into the inequality \(3x - y > 5\). This gives \(3(-1) - (-5) > 5\), which simplifies to \(-3 + 5 > 5\) or \(2 > 5\). This is false, so (-1,-5) is not a solution.
05

Substitute (1,-2) into the Inequality

For the pair (1,-2), substitute \(x = 1\) and \(y = -2\) into the inequality \(3x - y > 5\). This gives \(3(1) - (-2) > 5\), which simplifies to \(3 + 2 > 5\) or \(5 > 5\). This is not true, so (1,-2) is not a solution.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Ordered Pairs
An ordered pair consists of two numbers, written in the form \(x, y\). This notation is used to describe the coordinates of a point on a coordinate plane, where \(x\) represents the horizontal position and \(y\) the vertical position. Think of it as an address for each point in space. When dealing with inequalities such as \(3x - y > 5\), we analyze whether these pairs fit, or "satisfy", the conditions stated by the inequality.
For example, the ordered pair \(4, -1\) means \(x = 4\) and \(y = -1\). To check if this pair satisfies the inequality, we substitute these values into the expression.
Solutions to Inequalities
A solution to an inequality indicates which values make the inequality true. When analyzing each ordered pair in the context of an inequality, our goal is to determine whether substituting the numbers results in a true statement.
  • If the inequality is satisfied, then the ordered pair is a solution.
  • If the inequality is not satisfied, then it is not a solution.
For instance, looking at the inequality \(3x - y > 5\), let's say we take the pair \((4, -1)\). When substituted, if the resulting statement is true (e.g., \(13 > 5\)), then \((4, -1)\) is indeed a solution. Conversely, the pair \((0, 0)\) does not satisfy the inequality, meaning it isn't a solution. Thus, solutions to inequalities are the "addresses" that meet the conditions of the inequality.
Substitution Method
The substitution method is a reliable technique to determine if specific values satisfy an equation or inequality. It involves replacing variables with given values and simplifying the expression.
To utilize this method, follow these steps:
  • Take the inequality or equation you need to evaluate.
  • Replace the variables with the numbers from the ordered pair.
  • Simplify the expression to see if it results in a true statement.
Let's consider the inequality \(3x - y > 5\). For the ordered pair \(1, -2\), substitute \(x = 1\) and \(y = -2\), yielding the expression \(3(1) + 2 > 5\), which simplifies to \(5 > 5\). This expression is false, demonstrating that \((1, -2)\) is not a solution. By systematically applying substitution, we can verify which ordered pairs are solutions to a given inequality.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Decompose each rational expression into partial fractions using convenient values. $$\frac{-11 x+6}{5 x^{2}-4 x-12}$$

Market research has indicated that by \(2010,\) sales of MP3 portables will mushroom into a 70 billion dollar market. With a market this large, competition is often fierce- -with suppliers fighting to earn and hold market shares. For \(x\) million MP3 players sold, supply is modeled by \(y=10.5 x+25,\) where \(y\) is the current market price (in dollars). The related demand equation might be \(y=-5.20 x+140 .\) (a) How many million MP3 players will be supplied at a market price of 88 dollars ? What will the demand be at this price? Is supply less than demand? (b) How many million MP3 players will be supplied at a market price of 114 dollars ? What will the demand be at this price? Is demand less than supply? (c) To the nearest cent, at what price does the market reach equilibrium? How many units are being supplied/demanded?

Solve each system of inequalities by graphing the solution region. Verify the solution using a test point. $$\left\\{\begin{array}{c}x+2 y \geq 1 \\ 2 x-y \leq-2\end{array}\right.$$

Write each system as a matrix equation and solve (if possible) using inverse matrices and your calculator. If the coefficient matrix is singular, write no solution. $$\left\\{\begin{array}{l} 0.05 x-3.2 y=-15.8 \\ 0.02 x+2.4 y=12.08 \end{array}\right.$$

Decompose each rational expression into partial fractions using convenient values. $$\frac{x^{2}+24 x-12}{x^{3}-4 x}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.