Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 34

Chapter 2, Problem 34

Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 3-5(2x + 1) - 2(x-4) = 0

Verified step by step guidance

Start by distributing the constants outside the parentheses to each term inside the parentheses. For the expression $3 - 5(2x + 1) - 2(x - 4) = 0$, distribute $-5$ to both \$2x$ and \(1$, and distribute \)-2$ to both $x$ and $-4$.

Rewrite the equation after distribution: $3 - 10x - 5 - 2x + 8 = 0$.

Combine like terms on the left side of the equation. Group the constant terms together and the $x$ terms together.

Simplify the equation to the form $ax + b = 0$, where $a$ and $b$ are constants.

Solve for $x$ by isolating the variable: subtract or add constants to both sides and then divide both sides by the coefficient of $x$. After finding the solution, determine if the equation is an identity (true for all $x$), conditional (true for specific $x$), or inconsistent (no solution).

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Key Concepts

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

Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, and division. The goal is to find the value of the variable that makes the equation true.

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses, expressed as a(b + c) = ab + ac. This property is essential for simplifying expressions before solving equations.

Types of Equations: Identity, Conditional, and Inconsistent

An identity is true for all variable values, a conditional equation is true for specific values, and an inconsistent equation has no solution. Classifying the equation after solving helps understand the nature of its solutions.

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Related Practice

Textbook Question

In Exercises 29–36, simplify and write the result in standard form. √(1² - 4 × 0.5 × 5)

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Textbook Question

Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 3x/5 - x = x/10 - 5/2

Textbook Question

Solve each equation in Exercises 15–34 by the square root property. (2x + 8)² = 27

Textbook Question

For an international telephone call, a telephone company charges \$0.43 for the first minute, \$0.32 for each additional minute, and a \$2.10 service charge. If the cost of a call is \$5.73, how long did the person talk?

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Textbook Question

A job pays an annual salary of \$57,900, which includes a holiday bonus of \$1500. If paychecks are issued twice a month, what is the gross amount for each paycheck?

Textbook Question

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(x + 1) + 2 ≥ 3x + 6