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Chapter 5: Problem 111

Simplify. \(4 a+11-(-2 a-9)\)

Short Answer

Expert verified
The simplified expression is \(6a + 20\).

Step by step solution

01

Distribute the negative sign

Distribute the negative sign across the expressions inside the parentheses. This changes the sign of each term inside the parentheses: o\textbf{Expression:} \(4a + 11 - (-2a - 9)\). o\(4a + 11 + 2a + 9\)
02

Combine like terms

Combine terms with \(a\) and constant terms to simplify the expression: oCombine \(4a\) and \(2a\): \(6a\)oCombine \(11\) and \(9\): \(20\)oThus, the expression simplifies to \(6a + 20\)

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Key Concepts

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

distributive property
The distributive property is an essential algebraic tool that allows us to distribute a term across an expression inside parentheses. This property can make complex expressions simpler and easier to work with.

In our example, we begin with the expression: \(4a + 11 - (-2a - 9)\).

When we encounter a negative sign outside parentheses, we distribute it to every term inside. This means we flip the signs of each term inside. Hence, \(-(-2a)\) becomes \(+2a\) and \(-(-9)\) becomes \(+9\). After distributing the negative sign, the expression looks like this: \(4a + 11 + 2a + 9\).

Mastering the distributive property will help you tremendously as you advance in algebra!
combining like terms
Combining like terms is a key step in simplifying algebraic expressions. 'Like terms' are terms that have the same variable raised to the same power.

For example, in the expression \(4a + 11 + 2a + 9\), we can identify two groups of like terms:
  • The terms with the variable \(a\): \(4a\) and \(2a\)
  • The constant terms: \(11\) and \(9\)
To combine these like terms, we simply add or subtract their coefficients. For \(4a\) and \(2a\), we add their coefficients (\(4 + 2\)) to get \(6a\).
For the constants, add \(11 + 9\) to get \(20\).

Thus, combining like terms in our example gives us the simplified expression: \(6a + 20\).

This step consolidates terms, making the expression simpler and more compact.
integer operations
Integer operations involve the basic arithmetic operations鈥攁ddition, subtraction, multiplication, and division鈥攑erformed on integers.

In our example, handling integer operations is crucial when distributing signs and combining terms. Consider the operation when distributing the negative sign: \(-(-2a)\) turns into \(+2a\), and \(-(-9)\) turns into \(+9\).

The key rules to remember are:
  • Subtracting a negative is the same as addition.
  • Adding two positive integers gives a positive result.
  • Adding a positive integer to a negative integer (or vice versa) will depend on the magnitude of integers.
After combining like terms, we performed addition on integers: \(4 + 2\) for the terms with \(a\) and \(11 + 9\) for the constants.

By mastering integer operations, you will handle expressions and equations accurately and with ease.

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Most popular questions from this chapter

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=2 x^{3}\\\ &g(x)=5-x \end{aligned} $$

A grain of sand is placed on the first square of a chessboard, two grains on the second square, four grains on the third, eight on the fourth, and so on. Without a calculator, use scientific notation to approximate the number of grains of sand required for the 64 th square. (Hint: Use the fact that \(\left.2^{10} \approx 10^{3} \cdot\right)\)

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{1}{2-x}\\\ &g(x)=7-x \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=9-x^{2}\\\ &g(x)=\frac{3}{x-6}+2 x \end{aligned} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -16+5 $$
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