Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = -2|x|
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 22
Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 7x + 13 = 2(2x-5) + 3x + 23
Verified step by step guidance1
Start by expanding the right-hand side of the equation \(7x + 13 = 2(2x - 5) + 3x + 23\). Use the distributive property to multiply \(2\) by each term inside the parentheses: \(2 \times 2x\) and \(2 \times (-5)\).
Rewrite the equation after distribution: \(7x + 13 = 4x - 10 + 3x + 23\). Next, combine like terms on the right-hand side: combine \$4x\( and \)3x\(, and combine \)-10$ and \(23\).
After combining like terms, the equation becomes \(7x + 13 = 7x + 13\). Now, subtract \$7x$ from both sides to isolate the constants and variables separately.
Simplify the resulting equation after subtraction. You should get an equation involving only constants. Analyze this simplified equation to determine if it is always true, sometimes true, or never true.
Based on the simplified equation, classify the original equation as an identity (true for all values of \(x\)), a conditional equation (true for specific values of \(x\)), or an inconsistent equation (no solution).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Linear Equations
Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, and division. The goal is to find the value of the variable that makes the equation true.
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Distributive Property
The distributive property allows you to multiply a single term by each term inside parentheses, transforming expressions like a(b + c) into ab + ac. This step is essential for simplifying and solving equations involving parentheses.
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Types of Equations: Identity, Conditional, and Inconsistent
An identity is true for all variable values, a conditional equation is true for specific values, and an inconsistent equation has no solution. Identifying the type depends on the solution set after solving the equation.
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Related Practice
Textbook Question
You invested \$30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was \$705.88, how much was invested at each rate?
Textbook Question
In Exercises 15–26, use graphs to find each set. (- ∞, 5) ⋃ [1, 8)
Textbook Question
In Exercises 1–26, solve and check each linear equation. 2 - (7x + 5) = 13 - 3x
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Textbook Question
In Exercises 21–28, divide and express the result in standard form. 2/(3 - i)
Textbook Question
Solve each equation in Exercises 15–34 by the square root property. (x + 2)2 = 25