Problem 93 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$0.6 x-2(0.5 x+0.2) \leq 0.4-0.3 x$$

Step-by-Step Solution

Answer

The solution is \([-8, \infty)\).

Step 1: Distribute and Simplify

Start by distributing the expression inside the parentheses on the left side of the inequality. The expression \(-2(0.5x + 0.2)\) simplifies to \(-1x - 0.4\). Substitute this into the original inequality:\[0.6x - 1x - 0.4 \leq 0.4 - 0.3x\].Combining like terms, the inequality becomes:\[-0.4x - 0.4 \leq 0.4 - 0.3x\].

Step 2: Move All Terms to One Side

To isolate the variable \(x\), move all terms containing \(x\) to one side and constant terms to the other side. Add \(0.3x\) to both sides of the inequality:\[-0.4x + 0.3x - 0.4 \leq 0.4\].Simplifying gives:\[-0.1x - 0.4 \leq 0.4\].

Step 3: Isolate the Variable

Add \(0.4\) to both sides to get:\[-0.1x \leq 0.4 + 0.4\].This simplifies to:\[-0.1x \leq 0.8\].

Step 4: Solve for x

Divide both sides by \(-0.1\). Remember, when dividing or multiplying an inequality by a negative number, you must reverse the inequality sign:\[x \geq -8\].

Step 5: Write in Interval Notation

The inequality \(x \geq -8\) corresponds to the interval \([-8, \infty)\). This is the solution set in interval notation.

Step 6: Graphical Representation

To graph the solution \(x \geq -8\), draw a number line. Mark -8 on the line and shade to the right, indicating all numbers greater than or equal to -8. Use a closed dot on -8 because it is included in the solution set.

Key Concepts

Interval Notation

Interval notation is a way to describe the solution set of an inequality. It uses intervals to show a range of values that satisfy the condition. For the inequality solution \(x \geq -8\), it means all values of \(x\) starting from \(-8\) to positive infinity are valid solutions.

• A closed interval, like \([-8, \infty)\), includes the endpoint \(-8\), represented by a square bracket \([\).
• An open interval, signified by parentheses, excludes the endpoint.
• The use of infinity (\(\infty\)) always comes with open brackets because infinity itself isn't a precise number you can reach.

By writing the solution as \([-8, \infty)\), you concisely state that \(x\) starts at \(-8\) and includes every number greater than it.

Graphical Representation

Graphical representation provides a visual picture of an inequality solution. It's essential for understanding the range of solutions in a more intuitive way. For the inequality \(x \geq -8\), here’s how to represent it graphically:

• Draw a horizontal number line to serve as the base.
• Locate and mark \(-8\) on this line.
• Use a closed dot on \(-8\) to indicate that it is included in the solution set (because \(\geq\) includes equality).
• Shade the line to the right of \(-8\) to show that all numbers greater than \(-8\) are included.

This shaded part helps you quickly see that any number to the right of \(-8\) (and including \(-8\) itself) are solutions to the inequality.

Distribute and Simplify

Distributing and simplifying is a key skill for solving inequalities. When working with expressions, especially those that include parentheses, it’s important to distribute terms to simplify. In the original problem, you encounter the term \(-2(0.5x + 0.2)\).

• Distribute by multiplying \(-2\) with each term inside the parentheses.
• This process results in the expression \(-1x - 0.4\).
• Replace the original term with the simplified version to rewrite the inequality.

Once distribution is complete, combine like terms and simplify further. In algebra, these steps are crucial as they prepare the inequality for isolating the variable, making it easier to solve. Every simplification step helps reduce complexity, paving the way for finding solutions efficiently.