Problem 61 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(7(x+1)-8(x-2)<0\)

Step-by-Step Solution

Answer

(23, \infty)

Step 1: Distribute Constants

Expand the terms by distributing constants inside the parentheses. For the left side of the inequality, distribute 7 through (x+1) to get \(7x + 7\), and distribute -8 through (x-2) to get \(-8x + 16\). The inequality now looks like this: \[7x + 7 - 8x + 16 < 0\]

Step 2: Combine Like Terms

Combine like terms on the left side of the inequality. The terms involving \(x\) are \(7x\) and \(-8x\), which combine to \(-x\). The constant terms are \(7\) and \(16\), which add up to \(23\). Our inequality now becomes: \[-x + 23 < 0\]

Step 3: Isolate the Variable

Rearrange the inequality to solve for \(x\). Start by subtracting 23 from both sides:\[-x < -23\]To isolate \(x\), divide both sides by -1. Remember that dividing both sides of an inequality by a negative number reverses the inequality symbol. This gives:\[x > 23\]

Step 4: Express the Solution in Interval Notation

Since \(x > 23\), it means \(x\) can be any number greater than 23, but not equal to 23. In interval notation, this is expressed as:\((23, \infty)\)

Key Concepts

Interval Notation

Interval notation is a mathematical expression used to describe a set of numbers within a certain range. It is a concise way to represent continuous sets without using the traditional inequality symbols.

- Interval notation uses parentheses - Parentheses ")" or "(" mean the number is not included in the interval (open interval)- Brackets "]" or "[" mean the number is included in the interval (closed interval)

For example, the solution to the inequality where all numbers greater than 23 are valid, is written as \((23, \infty)\). This tells us that every number in the interval starting just after 23 and extending to infinity is part of the solution. Using interval notation helps:

• Avoid cumbersome inequality symbols
• Clearly indicate whether endpoint values are included or excluded
• Provide a standardized way to represent solution sets

Learning to use interval notation effectively will streamline your work in math, especially when dealing with inequalities.

Distribute Constants

Distributing constants in expressions involves multiplying a single constant by each term inside a set of parentheses. This process is also known as using the distributive property, which is a fundamental skill in algebra.

Let's break down our example:

• The expression is: \(7(x + 1) - 8(x - 2)\)
• Observe that 7 and -8 are constants placed in front of the parentheses.
• The term \(7(x + 1)\) becomes \(7x + 7\) after we distribute 7.
• The term \(-8(x - 2)\) simplifies to \(-8x + 16\) after distributing -8.

The distributive process ensures that each part within the parentheses is appropriately multiplied or combined. Remember:

• Distribute each constant to all terms inside the parentheses.
• This step prepares the expression for further simplification in the solving process.

By mastering the ability to distribute constants, you make complicated expressions simpler to work with.

Combine Like Terms

Combining like terms is the process of simplifying expressions by adding or subtracting terms that have the same variable part. This technique allows you to streamline expressions, making them easier to analyze and solve.

In our example, after distributing, we have:

• \(7x - 8x\) which are like terms and combine to form \(-x\).
• The constant terms are \(7 + 16\), summing up to \(23\).

The expression simplifies to \(-x + 23\). Why do we do this?

• Combining like terms reduces complexity, making the equation simpler.
• This step often leads directly to isolating the variable or solving the equation.

Always look for terms with identical variables and exponents to combine them effectively. This will keep your math work tidy and enable you to focus on solving the core problem.

Isolate Variable

Isolating the variable in an equation or inequality is critical to determining its solution. It involves manipulating the equation to get the variable by itself on one side.

Let's walk through this with our simplified inequality:

• Start with \(-x + 23 < 0\).
• Subtract 23 from both sides to obtain \(-x < -23\).
• In this situation where you need to solve for \(x\), divide every term by -1.
• Remember: dividing by a negative number reverses the inequality sign, resulting in \(x > 23\).

Crucially:

• Keeping the variable isolated often lets you see the solution or inequality clearly.
• Be cautious with signs, especially when multiplying or dividing by negatives.

Integrating these steps ensures clarity when solving inequalities and preparing to use interval notation for your solution.