Textbook Question
In Exercises 45–47, solve each formula for the specified variable. T = (A-P)/Pr for P
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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 47
In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 4(3x - 2) - 3x < 3(1 + 3x) - 7
Verified step by step guidance1
Start by expanding both sides of the inequality: expand \(4(3x - 2)\) and \(3(1 + 3x)\) to remove the parentheses.
After expansion, combine like terms on each side to simplify the inequality.
Next, get all terms containing \(x\) on one side of the inequality and constant terms on the other side by adding or subtracting terms accordingly.
Isolate the variable \(x\) by dividing or multiplying both sides of the inequality by the coefficient of \(x\). Remember, if you multiply or divide by a negative number, you must reverse the inequality sign.
Express the solution set in interval notation and then graph the solution on a number line, showing all values of \(x\) that satisfy the inequality.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Linear Inequalities
Linear inequalities involve expressions with variables raised to the first power and inequality symbols like <, >, ≤, or ≥. Solving them requires isolating the variable on one side by performing algebraic operations, similar to solving linear equations, but with attention to inequality rules.
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Properties of Inequalities
When solving inequalities, adding or subtracting the same number on both sides preserves the inequality direction. However, multiplying or dividing both sides by a negative number reverses the inequality sign. Understanding these properties is essential to maintain correct solution sets.
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Interval Notation and Graphing Solutions
Interval notation expresses solution sets as intervals on the number line, using parentheses for strict inequalities and brackets for inclusive inequalities. Graphing these solutions visually represents the range of values satisfying the inequality, aiding in interpretation and communication.
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Related Practice
Textbook Question
Solve each equation in Exercises 41–60 by making an appropriate substitution. x(-2) - x(-1) - 6 = 0
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Textbook Question
Write each English sentence as an equation in two variables. Then graph the equation. The y-value is the difference between four and twice the x-value.
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Textbook Question
Write each English sentence as an equation in two variables. Then graph the equation. The y-value is four more than twice the x-value.
Textbook Question
In Exercises 35–54, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? S = P + Prt for r
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Textbook Question
Perform the indicated operations and write the result in standard form.