Problem 20 - Algebra 2 and Trigonometry

Question

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ 14 c > 80-6 c $$

Step-by-Step Solution

Answer

The solution is \( c > 4 \).

Step 1: Combine like terms

Start by getting all the terms with the variable on one side of the inequality. Add \(6c\) to both sides to get:\[ 14c + 6c > 80 \]This simplifies to:\[ 20c > 80 \]

Step 2: Isolate the variable

Now, isolate the variable by dividing both sides of the inequality by 20, which is the coefficient of \(c\):\[ c > \frac{80}{20} \]This simplifies to:\[ c > 4 \]

Key Concepts

Integer Solutions

An integer solution refers to all the possible integer values that satisfy an inequality. In the given problem, we're looking for integer values of \(c\) that fulfill the inequality \(c > 4\).
This means any integer greater than 4 can be a solution. Consider the following points:

• The smallest integer that satisfies \(c > 4\) is 5.
• All integers greater than or equal to 5 also satisfy this inequation: 6, 7, 8, etc.
• Negative numbers and fractions between integers do not count as integer solutions for this inequality.

The understanding of integer solutions is crucial in inequalities since it helps you to distinguish between exact solutions and a range of possible values. This distinction is vital when solving practical problems that require whole numbers.

Combine Like Terms

Combining like terms is a crucial step in simplifying equations or inequalities. It involves consolidating terms with the same variable on one side. Let's break down our example:
The inequality starts with \(14c > 80 - 6c\).
To simplify, combine all terms involving \(c\) to one side. Add \(6c\) to both sides:

• First, identify terms containing the variable: on one side we have \(14c\), and on the other, there's \(-6c\).
• Add \(6c\) to both sides to combine the terms: \(14c + 6c > 80\).
• This simplifies to \(20c > 80\).

Combining like terms effectively reduces the complexity of equations or inequalities, making it easier to solve the problem during the next steps. It sets the stage to easily isolate the variable, which is often the following step in such problems.

Isolate the Variable

To isolate the variable in an inequality or equation means to get the variable on one side alone, with all other terms on the opposite side.In our example, the goal is to find the value of \(c\) such that \(20c > 80\). This means performing actions that will leave \(c\) on one side and make the inequality as simple as possible. Here's how it's done:

• Divide each side by the coefficient of the variable: in this case, divide both sides by 20, giving: \(c > \frac{80}{20}\).
• Simplify the fraction to get \(c > 4\).

Remember, when you divide or multiply both sides of an inequality, the direction of the inequality does not change, provided you did not divide by a negative number (which is not the case here). Isolating the variable helps in understanding what range of numbers the variable \(c\) can take, providing a clear solution for the inequality.