Problem 58 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ -6 x+2<-3(x+4) $$

Step-by-Step Solution

Answer

The solution is \(\{x \mid x > \frac{14}{3}\}\).

Step 1: Expand the Right Side

First, simplify the right-hand side of the inequality \(-3(x+4)\) using the distributive property. Multiply \(-3\) by each term inside the parentheses:\[-3(x+4) = -3 \cdot x - 3 \cdot 4 = -3x - 12\] The inequality now becomes:\[-6x + 2 < -3x - 12\]

Step 2: Add \(3x\) to Both Sides

To isolate the variable \(x\), start by eliminating \(-3x\) from the right side. Add \(3x\) to both sides:\[-6x + 3x + 2 < -3x + 3x - 12\]This simplifies to:\[-3x + 2 < -12\]

Step 3: Subtract 2 from Both Sides

Remove the constant \(2\) from the left side by subtracting \(2\) from both sides:\[-3x + 2 - 2 < -12 - 2\]This results in:\[-3x < -14\]

Step 4: Divide by \(-3\) and Reverse the Inequality

Since we are dividing by a negative number, the inequality sign must be reversed. Divide both sides by \(-3\):\[x > \frac{-14}{-3}\]This simplifies to:\[x > \frac{14}{3}\]

Step 5: Write the Solution in Set Notation

Express the solution using set notation, indicating the range of values for \(x\):\[\{x \mid x > \frac{14}{3}\}\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra that allows for the multiplication of a single term across a sum or difference within parentheses. It's commonly expressed as \(a(b+c) = ab + ac\). This rule applies universally, ensuring that any term outside the parentheses is distributed to each term inside.
For instance, in the inequality problem from our exercise, we apply the distributive property to \(-3(x+4)\). Here, \(-3\) needs to be multiplied by both \(x\) and \(4\), yielding \(-3x - 12\).

• Multiply \(-3\) by \(x\)
• Multiply \(-3\) by \(4\)

This simplifies the expression and helps to rearrange terms, crucial for solving inequality where isolating variables is required.

Solution Set Notation

Solution set notation is a mathematical style used to represent all possible solutions to an equation or inequality. It's typically represented in set-builder notation as \(\{ x \mid \, \text{condition} \}\). This simply means "the set of all \(x\) such that \(x\) meets the given condition."
In our exercise, after solving the inequality, the solution set is written as \(\{x \mid x > \frac{14}{3}\}\). This means that any value greater than \(\frac{14}{3}\) is a solution to the inequality.
This notation helps to concisely capture the range or specific solutions without listing every individual solution. It becomes particularly useful when dealing with inequalities that have infinite numbers of solutions.

Isolate the Variable

Isolating the variable refers to the process of manipulating an equation or inequality to get the variable in question, usually \(x\), by itself on one side. This is crucial because it allows us to determine the value or range of values that the variable can take.
In our example, isolating \(x\) involved several steps:

• Adding or subtracting terms to both sides to eliminate unwanted terms on the variable's side.
• Dividing by the coefficient of the variable.

Initially, we add \(3x\) to both sides to bring all \(x\) terms to one side, resulting in \(-3x + 2 < -12\). Next, by subtracting \(2\) from both sides, we further simplify to \(-3x < -14\).
Finally, dividing by \(-3\) yields \(x > \frac{14}{3}\), remembering that dividing by a negative number reverses the inequality sign. This methodical isolation of the variable allows clear determination of the solution set.