Domanda del libro di testo
Solve each radical equation in Exercises 88–89. √ (2x-3) + x = 3
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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 89
In Exercises 59–94, solve each absolute value inequality. 1 < |2 - 3x|
Guida verificata passo dopo passo1
Recall that the inequality \(1 < |2 - 3x|\) means the distance between \(2 - 3x\) and 0 is greater than 1.
Rewrite the inequality \(1 < |2 - 3x|\) as two separate inequalities: \(2 - 3x < -1\) or \(2 - 3x > 1\).
Solve the first inequality \(2 - 3x < -1\) by isolating \(x\): subtract 2 from both sides to get \(-3x < -3\), then divide both sides by \(-3\) (remember to reverse the inequality sign when dividing by a negative number).
Solve the second inequality \(2 - 3x > 1\) by isolating \(x\): subtract 2 from both sides to get \(-3x > -1\), then divide both sides by \(-3\) (again, reverse the inequality sign).
Combine the solutions from both inequalities to express the solution set for \(x\) that satisfies \(1 < |2 - 3x|\).
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Absolute Value Definition
The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A|, it equals A if A is non-negative, and -A if A is negative. Understanding this helps in rewriting absolute value inequalities into equivalent compound inequalities.
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Vertex Form
Solving Absolute Value Inequalities
To solve inequalities involving absolute values, such as |A| > c, we split them into two separate inequalities: A > c or A < -c when c is positive. This approach transforms the absolute value inequality into a compound inequality that can be solved using standard algebraic methods.
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Compound Inequalities
Compound inequalities involve two inequalities combined by 'and' or 'or'. For absolute value inequalities, solutions often form two intervals combined by 'or'. Understanding how to solve and graph compound inequalities is essential for interpreting the solution set correctly.
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