Problem 30 For Problems $1-50$, solve eac... [FREE SOLUTION]

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Elementary Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 3: Problem 30

For Problems $1-50$, solve each inequality. (Objectives 1 and 2) $$ 4(n-5)-2(n-1)<13 $$

Short Answer

Expert verified

The solution is $n < 15.5$.

Step by step solution

Distribute the Constants

Apply the distributive property to both terms on the left-hand side of the inequality. \[4(n-5) = 4n - 20\] \[-2(n-1) = -2n + 2\] So the inequality becomes: \[4n - 20 - 2n + 2 < 13\]

Combine Like Terms

Combine the like terms (terms with $n$) on the left-hand side.\[(4n - 2n) - 20 + 2 < 13\] \[2n - 18 < 13\]

Isolate the Variable Term

Add 18 to both sides of the inequality to isolate the term with $n$.\[2n - 18 + 18 < 13 + 18\] \[2n < 31\]

Solve for the Variable

Divide both sides by 2 to solve for $n$.\[\frac{2n}{2} < \frac{31}{2}\] \[n < 15.5\]

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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 a fundamental concept used in many areas of algebra. It helps in expanding expressions. To "distribute" means to multiply a single term with each of the terms within a set of parentheses.
This property is expressed by the formula:

Distribute: $ a(b + c) = ab + ac $

For example, in the inequality $4(n-5)-2(n-1)<13$, we use the distributive property to simplify each term containing a variable inside parentheses:

For $4(n-5)$, distribute 4 to get $4n - 20$.
For $-2(n-1)$, distribute $-2$ to obtain $-2n + 2$.

After distribution, the equation is easier to work with, changing it to: $4n - 20 - 2n + 2 < 13$. This crucial step lays the groundwork for solving the inequality.

Combining Like Terms

Once you've used the distributive property, the next step in solving equations and inequalities is to combine like terms. "Like terms" are terms that have the same variable raised to the same power. In simple terms, they look similar and can be combined by simple addition or subtraction.
So in the equation $4n - 20 - 2n + 2 < 13$, you should first identify which terms are alike:

Terms with $n$: $4n$ and $-2n$
Constant terms: $-20$ and $+2$

By combining the $n$ terms $(4n - 2n)$, you get $2n$. Similarly, combining the constants $(-20 + 2)$, you get $-18$.
This simplifies the inequality to: $2n - 18 < 13$. Combining like terms helps to reduce the complexity of the expression, making it easier to isolate the variable.

Isolation of Variables

After you have simplified the expression by distributing and combining like terms, the final step is to isolate the variable. This simply means getting the variable on one side of the equation or inequality by itself. This process often requires several steps.
In the inequality $2n - 18 < 13$, we aim to isolate $n$. Start by removing the constant $-18$ by adding 18 to both sides of the inequality:

$2n - 18 + 18 < 13 + 18$
This simplifies to $2n < 31$.

Next, isolate $n$ by dividing every term by 2:

$\frac{2n}{2} < \frac{31}{2}$
This results in $n < 15.5$.

Isolating the variable allows us to find the solution to the inequality, which tells us all possible values of $n$ that satisfy the original equation.

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91影视

For Problems \(1-50\), solve each inequality. (Objectives 1 and 2) $$ 4(n-5)-2(n-1)<13 $$

Short Answer

Step by step solution

Distribute the Constants

Combine Like Terms

Isolate the Variable Term

Solve for the Variable

Key Concepts

Distributive Property

Combining Like Terms

Isolation of Variables

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