Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 75

Chapter 2, Problem 75

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

Verified step by step guidance

Start by writing down the given equation: $10x + 3 = 8x + 3$.

To isolate the variable terms on one side, subtract \$8x$ from both sides: $10x + 3 - 8x = 8x + 3 - 8x$, which simplifies to $2x + 3 = 3$.

Next, subtract 3 from both sides to isolate the term with $x$: $2x + 3 - 3 = 3 - 3$, which simplifies to $2x = 0$.

Now, solve for $x$ by dividing both sides by 2: $\frac{2x}{2} = \frac{0}{2}$, which simplifies to $x = 0$.

Finally, analyze the solution: since you found a specific value for $x$, the equation is a conditional equation (true only for $x = 0$).

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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, or division. The goal is to find the value of the variable that makes the equation true.

Types of Equations: Identity, Conditional, and Inconsistent

An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solution. Identifying the type depends on the solution set after solving the equation.

Properties of Equality

Properties of equality, such as the addition, subtraction, multiplication, and division properties, allow manipulation of equations without changing their solutions. These properties are essential for maintaining balance while solving equations.

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Related Practice

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In Exercises 59–94, solve each absolute value inequality. |(2x + 2)/4| ≥ 2

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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair (2, 5) satisfies 3y - 2x = - 4.

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Solve each equation by the method of your choice. $3 x^{2} - 7 x + 1 = 0$

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In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4(x + 5) = 21 + 4x

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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

Exercises 73–75 will help you prepare for the material covered in the next section. Simplify: √18 - √8

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