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Chapter 0: Problem 88

Simplify each algebraic expression. $$2(5 x-1)+14 x$$

Short Answer

Expert verified
The simplified algebraic expression is \(24x - 2\).

Step by step solution

01

Apply the Distributive Property

Multiply 2 with each element inside the parentheses: \(2(5x) - 2(1)\) to get \(10x - 2\).
02

Combine Like Terms

Now that we've removed the parentheses, let's add together the like terms. We add \(10x - 2\) and \(14x\) to get \(24x - 2\).

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Key Concepts

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

Distributive Property
When faced with simplifying algebraic expressions involving parentheses, the distributive property is your go-to tool. This property allows us to eliminate parentheses by distributing the number outside across each term within. For example, in the expression \(2(5x - 1)\), the number 2 multiplies both 5x and -1. Let's break this down:
  • Multiply the number outside the parentheses with each term inside: \(2 \times 5x\) and \(2 \times -1\).
  • This results in \(10x - 2\).
Using the distributive property effectively opens the door to more straightforward expressions, setting the stage to combine like terms next. Always ensure you've applied multiplication to every term within the parentheses to avoid errors.
Combining Like Terms
After distributing, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. Their coefficients are the numbers in front of these variables and can be added or subtracted with each other. In the expression we simplified earlier, \(10x - 2 + 14x\), let's identify and combine like terms:
  • The terms \(10x\) and \(14x\) both have the variable \(x\), making them like terms.
  • Add their coefficients: \(10 + 14 = 24\).
  • The result is \(24x - 2\), with \(24x\) being the combined term.
Combining like terms tidies up your expression, making it cleaner and easier to interpret. Always look out for any remaining terms that can be combined at this stage.
Simplification of Expressions
The ultimate goal when working with algebraic expressions is simplification. This means expressing the equation in its simplest form, which involves performing both the distributive property and combining like terms until no further simplification is possible. The expression we鈥檝e been working with, \(2(5x - 1) + 14x\), simplifies to \(24x - 2\):
  • Each term inside the parentheses was multiplied by the factor outside, using the distributive property.
  • Like terms were identified and combined, resulting in \(24x\).
  • The constant term \(-2\) remains as is since there are no other constants to combine.
Simplifying is essential for solving equations and understanding their underlying structure. By simplifying, you're essentially preparing the expression for solving or further operation, ensuring clarity and accuracy in your calculations.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A local bank charges \(\$ 8\) per month plus 5垄 per check. The credit union charges \(\$ 2\) per month plus 8垄 per check. How many checks should be written each month to make the credit union a better deal?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(a x^{2}+c=0, a \neq 0,\) cannot be solved by the quadratic formula.

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

Perform the indicated operations. $$\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right)$$

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.
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