Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 76a

Chapter 2, Problem 76a

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

Verified step by step guidance

Start by isolating the variable term on one side of the equation. Subtract 2x from both sides of the equation:

5x + 7 - 2x = 2x + 7 - 2x

Simplify the equation by combining like terms:

(5x - 2x) + 7 = 7

, which simplifies to

3x + 7 = 7

Next, isolate the term with the variable by subtracting 7 from both sides:

3x + 7 - 7 = 7 - 7

Simplify the equation further:

3x = 0

Finally, solve for

x

by dividing both sides of the equation by 3:

x = \(\frac{0}{3}\)

. After solving, determine whether the equation is an identity, a conditional equation, or an inconsistent equation based on the solution.

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Key Concepts

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

Types of Equations

In algebra, equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solutions. Understanding these classifications helps in determining the nature of the solution set for any given equation.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. This typically includes combining like terms, using inverse operations, and simplifying both sides of the equation. For the equation 5x + 7 = 2x + 7, one would rearrange the terms to isolate x, which is essential for determining the type of equation it represents.

Checking Solutions

After solving an equation, it is crucial to check the solution by substituting it back into the original equation. This verification process confirms whether the solution is valid and helps identify the type of equation. For instance, if the left-hand side equals the right-hand side after substitution, it indicates whether the equation is an identity or a conditional equation.

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Related Practice

Textbook Question

In Exercises 59–94, solve each absolute value inequality. |3 - (2/3)x| > 5

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Textbook Question

Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

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Textbook Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. |3x - 1| = |x + 5|

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Textbook Question

In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. $x^{2} - 4 x - 5 = 0$

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Textbook Question

Solve each equation by the method of your choice. $(x - 3)^{2} - 25 = 0$

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Textbook Question

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 10x + 3 = 8x + 3

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