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
In Exercises 59–94, solve each absolute value inequality. |3 - (2/3)x| > 5
Guida verificata passo dopo passo1
Start by understanding that the inequality involves an absolute value expression: \(|3 - \frac{2}{3}x| > 5\). The absolute value inequality \(|A| > B\) means that either \(A > B\) or \(A < -B\).
Set up two separate inequalities based on the definition of absolute value inequalities:
1) \(3 - \frac{2}{3}x > 5\)
2) \(3 - \frac{2}{3}x < -5\)
Solve the first inequality \(3 - \frac{2}{3}x > 5\) by isolating \(x\):
- Subtract 3 from both sides: \(- \frac{2}{3}x > 2\)
- Multiply both sides by the reciprocal of \(-\frac{2}{3}\), which is \(-\frac{3}{2}\). Remember to reverse the inequality sign because you are multiplying by a negative number.
Solve the second inequality \(3 - \frac{2}{3}x < -5\) similarly:
- Subtract 3 from both sides: \(- \frac{2}{3}x < -8\)
- Multiply both sides by \(-\frac{3}{2}\), reversing the inequality sign again.
Combine the solutions from both inequalities to express the final solution set for \(x\). This will be the union of the two solution intervals obtained.
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Absolute Value Inequalities
An absolute value inequality involves expressions within absolute value bars and compares them to a number. To solve |A| > B, where B > 0, split it into two inequalities: A > B or A < -B. This approach helps find all values of the variable that satisfy the inequality.
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Linear Inequalities
Isolating the Variable
Before solving, isolate the variable expression inside the absolute value. This often involves simplifying the inequality and performing algebraic operations like addition, subtraction, multiplication, or division. Proper isolation ensures accurate splitting into two separate inequalities.
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05:28Equations with Two Variables
Solving Linear Inequalities
Once the absolute value inequality is split, solve each linear inequality separately. This involves applying standard inequality rules, such as reversing the inequality sign when multiplying or dividing by a negative number. The solution is the union of both inequality solutions.
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06:07Linear Inequalities
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