Problem 96 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$-0.9 x-(0.5+0.1 x)>-0.3 x-0.5$$

Step-by-Step Solution

Answer

Solution: \((-\infty, 0)\).

Step 1: Simplify the Inequality

First, simplify the left side of the inequality by distributing the negative sign over the terms inside the parenthesis. This gives:- \(-0.9x - 0.5 - 0.1x > -0.3x - 0.5\)

Step 2: Combine Like Terms

Next, combine the like terms on the left side of the inequality. The left side becomes:- \(-1.0x - 0.5 > -0.3x - 0.5\)

Step 3: Add 0.3x to Both Sides

To isolate the variable on one side, add \(0.3x\) to both sides:- \(-1.0x + 0.3x - 0.5 > -0.5\)- This simplifies to \(-0.7x - 0.5 > -0.5\).

Step 4: Add 0.5 to Both Sides

Add 0.5 to both sides of the inequality to further isolate the term with \(x\):- \(-0.7x > 0\)

Step 5: Divide by -0.7 and Reverse the Inequality

Divide each side of the inequality by \(-0.7\). Remember to reverse the inequality sign when you divide by a negative number:- \(x < 0\).

Step 6: Write the Solution in Interval Notation

The solution \(x < 0\) is expressed in interval notation as:- \((-\infty, 0)\).

Step 7: Graphical Representation of the Solution

To support the answer graphically, we plot this solution on a number line:- Draw a line and mark the point \(0\). Shade the region to the left of \(0\) and use an open circle at \(0\) to indicate that it is not included in the solution set.

Key Concepts

Interval Notation

Interval notation is a concise way of expressing solutions to inequalities. It shows which parts of the number line contain solutions. In our exercise, after solving the inequality \(x < 0\), we express the solution using interval notation. This is written as \((-fty, 0)\).
This tells us a few things about the solution:

• The interval starts at \(-\infty\), implying there is no lower bound; meaning all numbers less than any specific number are included.
• It ends at \(0\), indicating that \(0\) is the boundary point.
• The parenthesis \(()\) around \(-\infty\) and \(0\) imply that neither of these points is included in the solution set.

Interval notation complements the understanding of inequalities by showing the range of solutions and clarifying whether boundary points are included or excluded.

Graphical Representation

Graphical representation of inequalities helps visualize solutions on the number line, providing a clear depiction of which numbers satisfy the inequality.
For the solution \(x < 0\), here's how we graph it:

• Draw a horizontal line to represent the number line.
• Mark the number \(0\) distinctly on this line.
• Use an open circle at \(0\), which conveys that \(0\) itself is not part of the solution set.
• Shade the section of the line to the left of \(0\) to indicate all numbers less than \(0\) are included in our solution.

This graphical display aids in understanding how the inequality behaves and is particularly helpful in conveying complex mathematical solutions in an intuitive way.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations and inequalities to solve for a variable.
In the given problem \(-0.9x-(0.5+0.1x)>-0.3x-0.5\), solving it necessitated several algebraic steps:

• Initially, distribution was used on the left side of the inequality to eliminate parentheses: \(-0.9x - 0.5 - 0.1x\).
• Like terms were then combined to simplify both sides: \(-1.0x - 0.5\).
• The variable \(x\) was isolated by first adding \(0.3x\) to each side, leading us to \(-0.7x - 0.5 > -0.5\).
• Further simplification was achieved by adding \(0.5\) to both sides: \(-0.7x > 0\).
• The final step involved dividing by \(-0.7\), which required reversing the inequality sign, arriving at \(x < 0\).

These steps highlight key techniques in algebraic manipulation: distribution, combining like terms, isolation, and understanding the impact of operations, especially when involving inequalities.