Problem 51 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 3(x+2)-6>-2(x-3)+14 $$

Step-by-Step Solution

Answer

\(\{x | x > 4 \}\)

Step 1: Expand Both Sides

Begin by expanding both sides of the inequality. Distribute the numbers outside the parentheses: Left side: \(3(x+2) = 3x + 6\) Right side: \(-2(x-3) = -2x + 6\) The inequality becomes: \(3x + 6 - 6 > -2x + 6 + 14\)

Step 2: Simplify Both Sides

Simplify both sides by combining like terms. Left side simplifies to: \(3x\) (since \(+6\) and \(-6\) cancel each other out) Right side simplifies to: \(-2x + 20\) (\(+6\) and \(+14\) combine to \(+20\)) The inequality is now: \(3x > -2x + 20\)

Step 3: Move Variables to One Side

Move all terms with \(x\) to one side of the inequality. Add \(2x\) to both sides:\(3x + 2x > 20\) This simplifies to: \(5x > 20\)

Step 4: Solve for x

Divide both sides by 5 to isolate \(x\): \(x > \frac{20}{5}\) This simplifies to: \(x > 4\)

Step 5: Write Solution in Set Notation

Express the solution in set notation: The set of all \(x\) such that \(x > 4\) can be written as: \(\{x | x > 4 \}\)

Key Concepts

Solution Set Notation

When solving inequalities, expressing the solution using the correct notation is crucial. Solution set notation serves this exact purpose. It provides a neat way to showcase the range of solutions that satisfy the inequality. In solution set notation, we use curly brackets to indicate a set, and we express the condition inside.
For example, with the inequality solution like \(x > 4\), we write it as \(\{x \mid x > 4\}\). This notation essentially says, "the set of all \(x\) such that \(x\) is greater than 4." This form is neat, concise, and ideal for expressing solutions to inequalities.
Remember: the mid-bar \(\mid\) symbol reads as "such that," which helps clarify the condition satisfied by the numbers included in the set.

Expanding Equations

Expanding equations is a fundamental skill in algebra to simplify expressions. It involves distributing and removing parentheses. This process ensures you have a clear picture of all the terms you are dealing with.
Consider the inequality \(3(x+2)-6>-2(x-3)+14\). To expand, multiply each term inside the parentheses by the number outside:

• Left side: \(3(x+2)\) expands to \(3x + 6\).
• Right side: \(-2(x-3)\) expands to \(-2x + 6\).

This process simplifies the inequality into something more manageable, removing the parentheses and making it easier to see the next steps.

Combining Like Terms

After expanding, the next step in solving an inequality is combining like terms. This involves simplifying your expression by merging terms that have the same variable to power.

• For example, in the left side of \(3x + 6 - 6\), the \(+6\) and \(-6\) cancel each other out, resulting in just \(3x\).
• On the right side, \(-2x + 6 + 14\) simplifies to \(-2x + 20\) by adding the constants \(+6\) and \(+14\).

Combining like terms reduces the complexity of the expression, clearing the way to isolate the variable, which is your end goal in solving linear inequalities.

Linear Inequalities

Linear inequalities, similar to linear equations, feature a variable term and a constant term. However, instead of an equality, there’s an inequality sign such as \(>\), \(<\), \(\geq\), or \(\leq\). Solving linear inequalities involves finding the range of variable values that make the inequality true.
The problem asked for solving: \(3x > -2x + 20\).
To solve, we attempt to isolate the variable. First, we move \(-2x\) to the left side and simplify to get \(5x > 20\). Then, dividing both sides by 5 simplifies this to \(x > 4\).
Just as with equations, take care with operations involving inequality signs. Multiplying or dividing by a negative number flips the direction of the inequality sign. However, this does not occur in this problem as all operations maintain the initial inequality sign.