Domanda del libro di testo
In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 7 = 2x + 7
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 2, Problema 77
The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. |3x - 1| = |x + 5|
Guida verificata passo dopo passo1
Recognize that the equation |3x - 1| = |x + 5| can be rewritten using the property that if |u| = |v|, then u = v or u = -v. Here, let u = 3x - 1 and v = x + 5.
Set up the two separate equations based on the property: 3x - 1 = x + 5 and 3x - 1 = -(x + 5).
Solve the first equation 3x - 1 = x + 5 by isolating x: subtract x from both sides and add 1 to both sides to get 2x = 6, then solve for x.
Solve the second equation 3x - 1 = -x - 5 by distributing the negative sign, then combine like terms: 3x - 1 = -x - 5, add x to both sides and add 1 to both sides to get 4x = -4, then solve for x.
Check both solutions by substituting them back into the original equation to verify that the absolute values are equal.
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Absolute Value Definition
The absolute value of a number or expression represents its distance from zero on the number line, always yielding a non-negative result. For any expression u, |u| equals u if u is non-negative, and -u if u is negative. This concept is fundamental when solving equations involving absolute values.
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Vertex Form
Equating Absolute Values
When two absolute values are equal, |u| = |v|, it implies that either u = v or u = -v. This property allows us to rewrite the equation without absolute value bars by considering both positive and negative scenarios, which is essential for finding all possible solutions.
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06:00Categorizing Linear Equations
Solving Linear Equations
After rewriting the absolute value equation as two separate linear equations, solving each involves isolating the variable using algebraic operations like addition, subtraction, multiplication, or division. Understanding how to solve linear equations is crucial to find the values of x that satisfy the original equation.
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04:02Solving Linear Equations with Fractions
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