Problem 63 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(5(x-6)-6(x+2)<0\)

Step-by-Step Solution

Answer

The solution in interval notation is \((-42, \infty)\).

Step 1: Distribute the Constants

Start by distributing the constants inside the parentheses.For the expression \(5(x - 6)\), distribute the 5:\[5(x - 6) = 5x - 30\]For the expression \(-6(x + 2)\), distribute the -6:\[-6(x + 2) = -6x - 12\]The inequality becomes:\(5x - 30 - 6x - 12 < 0\)

Step 2: Combine Like Terms

Combine like terms on the left side of the inequality.Combine the \(5x\) and \(-6x\):\[5x - 6x = -x\]Combine the constants \(-30\) and \(-12\):\[-30 - 12 = -42\]The inequality now is:\(-x - 42 < 0\)

Step 3: Isolate the Variable Term

Add 42 to both sides to isolate the \(-x\) term:\[-x - 42 + 42 < 0 + 42\]Simplify:\[-x < 42\]

Step 4: Solve for x

Multiply both sides by \(-1\) to solve for \(x\), remembering to reverse the inequality sign:\[x > -42\]

Step 5: Write the Solution in Interval Notation

Express the solution \(x > -42\) in interval notation:\((-42, \infty)\)

Key Concepts

Step-by-step solutions

Step-by-step solutions are a powerful tool to help you fully understand how to solve algebraic inequalities. By breaking down the problem into manageable parts, these solutions guide you through each stage of the process.
This method ensures that you don't get lost or overwhelmed by complex calculations. Let's see how this approach works using this inequality:

• First, identify and execute necessary operations like distribution or combining like terms.
• Next, isolate the variable step-by-step, ensuring every arithmetic operation is clear and justified.
• Finally, transform your solution into its required form, like interval notation, at the end of the process.

At each step, it is essential to check and double-check your work to ensure accuracy. If every small element of the problem is understood, tackling similar problems becomes much more manageable.

Interval notation

Interval notation is a convenient way of expressing the solution set of an inequality. Once you've solved the inequality, it's crucial to understand how to present the solution correctly. Let's delve into how to write interval notation.To express the solution in interval notation for an inequality like \(x > -42\):

• The interval starts from the smallest value in the solution set, which is next to a parenthesis \((-42,\ldots\)
• Since \(x\) can take any value greater than \(-42\) but not including \(-42\), use the parenthesis on \(-42\) to indicate it's not part of the solution.
• If the solution continues infinitely, use \(\infty\) to show that it goes on without bound, written as \((..., \infty)\).

The use of \(\infty\) always requires a parenthesis because infinity is not a number you reach or include. This establishes a comprehensive way to communicate solutions of inequalities efficiently and compactly.

Distributive property

The distributive property is a fundamental algebraic tool that simplifies expressions by removing parentheses. It involves distributing a factor across terms within parentheses. Understanding this property is essential when solving inequalities like our example.Here's how the distributive property works with our inequality:- For the term \(5(x - 6)\), distribute the 5 across the terms inside the parenthesis to get \(5x - 30\).- Similarly, for \(-6(x + 2)\), distribute \(-6\) to yield \(-6x - 12\).The equation then simplifies to \(5x - 30 - 6x - 12\). This simplification is essential as it transforms the inequality into a form where you can easily combine like terms and further solve for the variable.
With practice, the application of the distributive property becomes second nature, helping to avoid errors and streamline solving processes.