Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 57

Chapter 2, Problem 57

In Exercises 51–58, solve each compound inequality. - 3 ≤ (2/3)x - 5 < - 1

Verified step by step guidance

Start by writing the compound inequality clearly: $-3 \leq \frac{2}{3}x - 5 < -1$.

Add 5 to all three parts of the inequality to isolate the term with $x$: $-3 + 5 \leq \frac{2}{3}x - 5 + 5 < -1 + 5$.

Simplify each part: $2 \leq \frac{2}{3}x < 4$.

Multiply all parts of the inequality by the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$, to solve for $x$: $2 \times \frac{3}{2} \leq x < 4 \times \frac{3}{2}$.

Simplify the multiplication to find the range of $x$: $3 \leq x < 6$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Compound Inequalities

Compound inequalities involve two inequalities joined together, often by 'and' or 'or'. In this problem, the compound inequality uses 'and', meaning the solution must satisfy both inequalities simultaneously. Understanding how to split and solve each part is essential.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable by performing inverse operations, similar to solving equations, but with attention to inequality direction. Multiplying or dividing by a negative number reverses the inequality sign, which is crucial to remember.

Interval Notation and Solution Sets

After solving inequalities, expressing the solution set in interval notation helps clearly communicate the range of values that satisfy the inequality. Understanding how to write and interpret intervals is important for representing solutions concisely.

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Interval Notation

Related Practice

Textbook Question

Solve each equation in Exercises 58–59 by factoring. 2x^2 +15x = 8

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. 3/(x + 2) + 2/(x - 2) = 8/(x + 2)(x - 2)

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Textbook Question

Write each power of i as i, - 1, - i, or 1.

i¹¹⁴

Textbook Question

Write each power of i as i, - 1, - i, or 1.

i⁴⁴

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 3 x - 1 = 0$

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Textbook Question

In Exercises 48–57, perform the indicated operations and write the result in standard form. √ (-32) - √ (-18)