Domanda del libro di testo
Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. (3x+1)/3 - 13/2 = (1-x)/4
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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 27
Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(2x + 3) + √(x - 2) = 2
Guida verificata passo dopo passo1
Identify the equation: \(\sqrt{2x + 3} + \sqrt{x - 2} = 2\). Our goal is to solve for \(x\).
Isolate one of the square root terms. For example, subtract \(\sqrt{x - 2}\) from both sides to get: \(\sqrt{2x + 3} = 2 - \sqrt{x - 2}\).
Square both sides of the equation to eliminate the square root on the left. This gives: \(\left(\sqrt{2x + 3}\right)^2 = \left(2 - \sqrt{x - 2}\right)^2\).
Simplify both sides: the left side becomes \(2x + 3\), and the right side expands using the formula \((a - b)^2 = a^2 - 2ab + b^2\) to \(4 - 4\sqrt{x - 2} + (x - 2)\).
Rearrange the equation to isolate the remaining square root term, then square both sides again to eliminate it. After that, solve the resulting polynomial equation for \(x\). Finally, check all proposed solutions in the original equation to discard any extraneous roots.
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Radical Equations
Radical equations involve variables within a root, typically a square root. Solving them requires isolating the radical expression and then eliminating the root by raising both sides of the equation to the appropriate power, often squaring. This process can introduce extraneous solutions, so checking all proposed solutions is essential.
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Percorso guidato
05:20
Expanding Radicals
Domain Restrictions
The domain of a radical equation is limited by the requirement that the expression inside the square root must be non-negative. For example, in √(2x + 3), the expression 2x + 3 must be ≥ 0. Identifying these restrictions helps determine valid solutions and avoid invalid or extraneous answers.
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Percorso guidato
3:51Domain Restrictions of Composed Functions
Checking Solutions
After solving a radical equation, substituting the solutions back into the original equation is crucial to verify their validity. Squaring both sides can introduce extraneous solutions that do not satisfy the original equation. Checking ensures only true solutions are accepted.
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Percorso guidato
05:21Restrictions on Rational Equations
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