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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 47
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

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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

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 5/2x - 8/9 = 1/18 - 1/3x

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Textbook Question

Use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.

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Textbook Question

Use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.

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Textbook Question

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. (x - 2)/2x + 1 = (x + 1)/x

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Textbook Question

Solve each equation in Exercises 47–64 by completing the square. x2+6x=7x^2 + 6x = 7

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Textbook Question

Perform the indicated operations and write the result in standard form. −6−−1248\(\frac{-6 - \sqrt{-12}\)}{48}

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