/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 86 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 86

Simplify each expression. $$ -2(x-3)-4(2 x+1)-x $$

Short Answer

Expert verified
-11x + 2

Step by step solution

01

Distribute the constants

Apply the distributive property to each term of the expression. For the first term, distribute -2 across ( x - 3 ): -2(x - 3) = -2x + 6 . For the second term, distribute -4 across (2x + 1): -4(2x + 1) = -8x - 4
02

Substitute back into the Expression

Substitute the distributed terms back into the original expression: -2x + 6 - 8x - 4 - x
03

Combine Like Terms

Group the like terms together and combine them: (-2x - 8x - x) + (6 - 4) = -11x + 2
04

Simplified Expression

The simplified expression is: -11x + 2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
To simplify algebraic expressions, understanding the Distributive Property is essential. This property shows how to multiply a constant by a sum or difference inside parentheses.

For instance, in the expression -2(x - 3), the -2 must be distributed to both x and -3. Performing the distribution, we get:

\(-2 \cdot x + -2 \cdot (-3) = -2x + 6\).

Similarly, for -4(2x + 1), the -4 multiplies both 2x and 1, giving:

\(-4 \cdot 2x + -4 \cdot 1 = -8x - 4\).

By distributing correctly, you ensure all parts of the expression are handled before moving on. This step transforms complicated expressions into simpler forms that are easier to manage.
Combining Like Terms
After applying the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power.

Consider the expression: \(-2x + 6 - 8x - 4 - x\).

Here, -2x, -8x, and -x are like terms because they all contain the variable x. To combine them, you simply add or subtract their coefficients:

\(-2x - 8x - x = -11x\).

Numbers without variables, like 6 and -4, are also like terms and can be combined:

\(6 - 4 = 2\).

By grouping and combining like terms, the expression becomes easier to interpret and solve.
Simplifying Expressions
Simplifying expressions is the final stage of solving algebraic problems. Once the distributive property has been utilized and like terms are combined, you aim to rewrite the expression in its simplest form.

For the given problem, after distributing and combining, you get: \(-11x + 2\).

This expression is now simplified because there are no more like terms to combine, and no further distribution is needed. Simplified expressions are often more manageable and can be more easily used in further calculations or problem-solving scenarios.

Remember, the goal of simplification is to reduce the expression to its simplest possible form, without changing its value.

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Most popular questions from this chapter

Solve each problem. Cooperative learning The sum of the integers from 1 through \(n\) is \(\frac{n(n+1)}{2}\) The sum of the squares of the integers from 1 through \(n\) is \(\frac{n(n+1)(2 n+1)}{6} .\) The sum of the cubes of the integers from 1 through \(n\) is \(\frac{n^{2}(n+1)^{2}}{4} .\) Use the appropriate expressions to find the following values. a) The sum of the integers from 1 through 50 b) The sum of the squares of the integers from 1 through 40 c) The sum of the cubes of the integers from 1 through 30 d) The square of the sum of the integers from 1 through 20 e) The cube of the sum of the integers from 1 through 10

Find each product or quotient. $$ \frac{2 x-4}{-2} $$

Simplify. $$ \frac{4 w y t}{4}+\frac{-8 w y t}{2}-\frac{-2 w y}{2} $$

Simplify each expression. $$ x^{2}-\left(x^{2}-y^{2}-z\right) $$

Simplify. $$ 3 a^{2} w^{2}-5 w^{2} \cdot a^{2}-2 a w \cdot 2 a w $$
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