Problem 32 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 12 x-4(3 x+5) \leq-2 $$

Step-by-Step Solution

Answer

No solution; the interval notation is \( \emptyset \).

Step 1: Distribute the Negative Sign and Simplify

First, distribute the negative 4 through the terms in the parenthesis to simplify the inequality.Given:\[ 12x - 4(3x + 5) \leq -2 \]Distribute:\[ 12x - 12x - 20 \leq -2 \]Combine like terms:\[ -20 \leq -2 \]

Step 2: Solve the Inequality

Now solve the simplified inequality.We have:\[ -20 \leq -2 \]Because this is a false statement, it indicates that there is no solution.

Step 3: Graph the Solution Set

Since the inequality is false, there are no values of \(x\) that satisfy the inequality, meaning no solution set to graph. The graph would be an empty set on the number line.

Step 4: Write the Solution in Interval Notation

Since there is no solution, the interval notation for the solution set is an empty set, which can be represented by \( \emptyset \).

Key Concepts

Distributive Property

When solving inequalities, one common step involves using the distributive property to simplify expressions. The distributive property states that for any numbers or expressions, if you have an expression like \(a(b + c)\), it can be expanded to \(ab + ac\).
In our exercise, we apply it to \(-4(3x + 5)\). This means:

• Multiply \(-4\) by \(3x\), resulting in \(-12x\).
• Next, multiply \(-4\) by \(5\), giving \(-20\).

By applying the distributive property, the inequality simplifies to \(12x - 12x - 20 \leq -2\). This step is crucial in setting the stage for solving the inequality.

Inequality Solution

Solving an inequality involves finding the values for which the inequality holds true. Usually, after simplifying the expression using the distributive property, you'll combine like terms to get a usable form of the inequality. In our case, the expression simplifies to \(-20 \leq -2\).
This statement is false because \(-20\) is not less than or equal to \(-2\). Thus, this inequality has no solutions.
When an inequality is false, it means there are no values of \(x\) that make the inequality true. This often happens when the inequality simplifies to a clearly untrue statement.

Interval Notation

Interval notation is a way of writing the set of solutions to an inequality using brackets and parentheses. It is a concise form to represent all numbers between two endpoints.
In scenarios where there is no solution, such as our example with \(-20 \leq -2\), the solution set is empty. In interval notation, an empty set is written as \(\emptyset\).
This indicates that there is no range of values that satisfy the inequality, and it is a clear and simple notation to express this fact.

Graphing Inequalities

Graphing inequalities typically involves marking a number line with the solutions found from the inequality. The solutions are represented as intervals or specific points on this line.
However, in cases where the inequality has no solution, like our example, the graph remains unchanged and no values are shaded or marked. This is often represented by an empty number line or simply no markings at all.
Graphing allows a visual representation to easily see the range of solutions, but with no solution present, the graph is effectively empty. This visually shows that there are no values making \(-20 \leq -2\) true.