Problem 9 - Thomas Calculus

Question

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ 2 x-\frac{1}{2} \geq 7 x+\frac{7}{6} $$

Step-by-Step Solution

Answer

The inequality solution is \(x \leq -\frac{1}{3}\).

Step 1: Isolating Variable Terms

First, let's isolate the variable terms on one side of the inequality. Start by subtracting \(7x\) from both sides:\[ 2x - \frac{1}{2} - 7x \geq \frac{7}{6} \] This simplifies to:\[ -5x - \frac{1}{2} \geq \frac{7}{6} \]

Step 2: Isolating Constant Terms

Now, isolate the constant term by adding \(\frac{1}{2}\) to both sides of the inequality:\[ -5x - \frac{1}{2} + \frac{1}{2} \geq \frac{7}{6} + \frac{1}{2} \]This results in:\[ -5x \geq \frac{7}{6} + \frac{3}{6} \]which simplifies to:\[ -5x \geq \frac{10}{6} \]

Step 3: Solving for the Variable

Divide each side of the inequality by \(-5\) to solve for \(x\). Remember to reverse the inequality sign since you are dividing by a negative number:\[ x \leq -\frac{10}{6} \times \frac{1}{5} \]which simplifies to:\[ x \leq -\frac{1}{3} \]

Step 4: Graphing the Solution on a Real Line

The solution can be represented on a number line where all values less than or equal to \(-\frac{1}{3}\) are included. This is shown by shading the line to the left of \(-\frac{1}{3}\) and including a closed circle at \(-\frac{1}{3}\).

Key Concepts

Variable Isolation

Variable isolation is like sorting out a messy drawer; you want each item in its right place. In math, we often want to find the value of a variable, like \(x\), in an inequality. To do this, we take steps to "isolate" the variable on one side of the inequality. Think of this as clearing the path so that \(x\) stands alone.Here's how it works:

• First, we rearrange the inequality to get all terms with the variable \(x\) on one side and constant numbers on the other.
• In our example, we subtract \(7x\) from both sides which gives us \(-5x - \frac{1}{2} \geq \frac{7}{6}\).
• Then, move constants around by adding or subtracting. Add \(\frac{1}{2}\) to clean up the constants, and you get \(-5x \geq \frac{10}{6}\).

This process helps make the inequality clearer and prepares us to solve for \(x\) next.

Graphing Inequalities

Graphing inequalities helps us visually understand the solution. Imagine you are drawing a line and marking parts that are acceptable for the inequality.Here's what you do:

• First, solve the inequality thoroughly to know which numbers satisfy it. In the example, \(x \leq -\frac{1}{3}\) is what we found.
• On a number line, locate \(-\frac{1}{3}\). Place a closed circle here because \(x\) can actually be \(-\frac{1}{3}\).
• Shade to the left of \(-\frac{1}{3}\) showing that all numbers less than \(-\frac{1}{3}\) satisfy the inequality.

This shading and circling method on a number line quickly illustrates the solution set and is a simple yet powerful tool to visualize which numbers make the inequality true.

Solution Sets

Solution sets represent all the possible answers that satisfy an inequality. Once we solve an inequality, the solution set wraps up all these answers neatly.For example:

• The solution set for the inequality \(2x - \frac{1}{2} \geq 7x + \frac{7}{6}\) is \(x \leq -\frac{1}{3}\).
• This means any number less than or equal to \(-\frac{1}{3}\) will work in the original inequality.
• We express this on a number line, helping anyone understand the range of numbers at a glance.

Understanding solution sets is crucial because they tell us all the possibilities, not just a single number. It opens up the entire range of values that make an inequality true.