Problem 46 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 3(x-5)<2(2 x-1) $$

Step-by-Step Solution

Answer

\(\{ x | x > -13 \}\)

Step 1: Distribute on both sides

Start by distributing the numbers outside the parentheses on both sides of the inequality. For the left-hand side, distribute the 3 over \(x - 5\): \(3(x - 5) = 3x - 15\). For the right-hand side, distribute the 2 over \(2x - 1\): \(2(2x - 1) = 4x - 2\). This simplifies our inequality to: \(3x - 15 < 4x - 2\).

Step 2: Rearrange terms

Subtract \(3x\) from both sides of the inequality to move the variable terms to one side: \(3x - 15 - 3x < 4x - 2 - 3x\). This simplifies further to \(-15 < x - 2\).

Step 3: Isolate x

Add 2 to both sides of the inequality so that \(x\) is isolated: \(-15 + 2 < x - 2 + 2\). This results in \(-13 < x\).

Step 4: Write the solution set

Now that \(-13 < x\), rewrite the inequality in solution set notation. The solution set is \(\{ x | x > -13 \}\), which indicates that \(x\) can be any number greater than \(-13\).

Key Concepts

Solution Set Notation

When solving inequalities, it's important to express your answer clearly and concisely. Often, we use "solution set notation" to do this. This notation helps show the range of numbers that solve the inequality.

Let's break down the process. When you're given a solution like \(x > -13\), it means that any number greater than \(-13\) will satisfy the inequality. In solution set notation, this is expressed as \({ x | x > -13 }\).

This format uses curly brackets "{ }" to contain the expression, a vertical bar "|" to mean "such that" or "for which," and then the inequality condition that describes the solution set. Basically, you're saying "the set of all x such that x is greater than -13."

• Curly braces are symbols used to enclose the set.
• The vertical bar "|" translates to "such that."
• The expression to the right of the bar specifies the condition.

Using solution set notation makes it clear that we're dealing with a set of solutions and not just a single number.

Distributive Property

The distributive property is a fundamental concept used in algebra, especially when solving equations or inequalities that involve parentheses. It allows you to "distribute" a multiplying term over any terms inside parentheses.

For example, if you have an expression like \(a(b + c)\), the distributive property lets you rewrite it as \(ab + ac\). This means you multiply the outside term \(a\) by each term inside the parentheses separately and then add the results.

Let's apply this to the original exercise: \(3(x-5)<2(2x-1)\)

• On the left side, multiply 3 by each term in \(x - 5\), resulting in \(3x - 15\).
• On the right side, multiply 2 by each term in \(2x - 1\), leading to \(4x - 2\).

Using the distributive property is crucial for simplifying complex expressions and making them easier to work with.

Inequality Steps

Solving inequalities involves a series of strategic steps similar to solving equations, but with some additional considerations.

Here's how you can solve an inequality step by step:

• **Simplify both sides**: Use the distributive property if necessary to eliminate parentheses.
• **Rearrange terms**: Get all terms containing variables on one side and constant numbers on the other. This can involve adding or subtracting terms from both sides.
• **Isolate the variable**: You want the variable by itself on one side of the inequality. Do this by adding or subtracting numbers as needed.
• **Consider directionality**: When multiplying or dividing by a negative number, remember to flip the inequality sign.

Let's summarize how these steps apply to our exercise:- We first used the distributive property to simplify both sides, turning \(3(x - 5) < 2(2x - 1)\) into \(3x - 15 < 4x - 2\).- Next, we rearranged terms by subtracting \(3x\) from both sides, resulting in \(-15 < x - 2\).- Finally, isolating \(x\) by adding 2 to both sides gave us \(-13 < x\).
Having a clear process helps ensure that errors are minimized, and any complexity in the inequality is quickly resolved.