Problem 96 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The solution is \((-\infty, 0)\).

Step 1: Simplify Both Sides

Start by distributing and combining like terms on each side of the inequality.Given: \[-0.9x - (0.5 + 0.1x) > -0.3x - 0.5\]First, distribute the negative sign on the left side:\[-0.9x - 0.5 - 0.1x > -0.3x - 0.5\]Combine like terms:\[-(0.9x + 0.1x) - 0.5 > -0.3x - 0.5\]\[-1.0x - 0.5 > -0.3x - 0.5\]

Step 2: Add and Subtract Terms

Move all the terms involving \(x\) to one side and the constant terms to the other side.Add \(1.0x\) to both sides:\[-0.5 > 0.7x - 0.5\]Next, add \(0.5\) to both sides to isolate the term involving \(x\):\[0 > 0.7x\]

Step 3: Solve for x

Divide both sides by 0.7 to solve for \(x\).\[x < 0\]

Step 4: Write the Solution Set in Interval Notation

The solution set in interval notation is:\((-\infty, 0)\)

Step 5: Graph the Solution

On a number line, shade the region left of 0 (not including 0), because \(x\) can take any value less than 0 but not equal to 0. The endpoint at 0 is represented with an open circle.

Key Concepts

Interval Notation

Interval notation is a way to describe the set of solutions for an inequality. It uses parentheses and brackets to precisely denote the range of values. Parentheses, \((\underline{\phantom{xxx}})\), signify that an endpoint is not included in the interval, whereas brackets, \([\underline{\phantom{xxx}}]\), indicate inclusion.

For our inequality solution, \(x < 0\), we use interval notation to write \(( -\infty, 0) \). This notation communicates that all solution values for \(x\) are less than 0, extending indefinitely towards negative infinity.

• The \(-\infty\) suggests the solution stretches infinitely in the negative direction.
• The 0 has a parenthesis, \((0)\), meaning 0 itself is not part of the solution set.

Understanding interval notation is crucial as it concisely and effectively communicates the solution set for any inequality, making it clear and concise for mathematical communication.

Graphical Representation

A graphical representation on a number line visually conveys the solution to an inequality. Number lines help display which numbers satisfy the inequality's condition.

Here's how you can graph \(x < 0\):

• Draw a horizontal line and mark 0 on this line.
• Use an open circle at 0. An open circle indicates that, in this inequality, the value 0 is not included.
• Shade the area to the left of 0 to represent all numbers less than 0 satisfying the inequality.

This graphical approach supports our solution by providing a visual interpretation. It allows students to immediately see all possible values of \(x\) satisfying the inequality condition.

Solving Inequalities

Solving inequalities involves several key steps that enable us to find all possible solutions. Here's a simple breakdown of how to approach solving inequalities:

1. **Simplify Both Sides:** Begin by simplifying each side of the inequality separately—just like you would with an equation. Distribute any coefficients and combine like terms. In our example, we distributed and simplified to rewrite the initial inequality.2. **Isolate the Variable:** Start moving all terms involving the variable \(x\) to one side. For our case, this was done by adding 1.0\(x\) to each side and then simplifying further.3. **Solve for the Variable:** To isolate \(x\), we use inverse operations, just like solving regular equations. We added or subtracted terms to isolate terms with \(x\). Finally, dividing both sides by the coefficient of \(x\) completed solving the inequality.4. **Check Your Solution:** Always verify the solution by substituting some values from the solution back into the original inequality to ensure they make the inequality true.These steps create a structured path to finding the solution for any inequality. Perfecting each step leads to a solid understanding of solving inequalities systematically.