Problem 79 - A Graphical Approach to College Algebra

Question

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(1-2 x=0\) (b) \(1-2 x \leq 0\) (c) \(1-2 x \geq 0\)

Step-by-Step Solution

Answer

(a) \(x = \frac{1}{2}\), (b) \([\frac{1}{2}, \infty)\), (c) \((-\infty, \frac{1}{2}]\).

Step 1: Solving the Equation

The equation given is \(1 - 2x = 0\). We need to solve for \(x\). To do this, first add \(2x\) to both sides to get \(1 = 2x\). Next, divide both sides by 2 to isolate \(x\), yielding \(x = \frac{1}{2}\).

Step 2: Solving the Inequality \(1 - 2x \leq 0\)

To solve the inequality \(1 - 2x \leq 0\), follow similar steps as solving the equation. Start by adding \(2x\) to both sides, which gives \(1 \leq 2x\). Then divide both sides by 2 to isolate \(x\), resulting in \(\frac{1}{2} \leq x\). In interval notation, this solution is \([\frac{1}{2}, \infty)\).

Step 3: Solving the Inequality \(1 - 2x \geq 0\)

The inequality \(1 - 2x \geq 0\) is solved similarly. Add \(2x\) to both sides to get \(1 \geq 2x\). Divide both sides by 2, leading to \(\frac{1}{2} \geq x\). The interval notation for this solution is \(( -\infty, \frac{1}{2}]\).

Key Concepts

Interval Notation

Interval notation is a neat and efficient way of writing down the solution set of inequalities and showing the range of values for which a condition holds true. It tells us exactly which numbers satisfy an inequality and include details such as whether the endpoints are included or not included.

To break it down, interval notation uses brackets and parentheses to denote these ranges:

• Brackets [ ] are used if the endpoint is included, meaning the inequality includes \(\leq\) or \(\geq\).
• Parentheses ( ) are used if the endpoint is not included, meaning the inequality includes \(<\) or \(>\).

For example, in the inequality \(1 - 2x \leq 0\), we isolate \(x\) and find out the solution using dividing and rearranging steps. The solution becomes \([\frac{1}{2}, \infty)\), which tells us \(x\) starts from \(\frac{1}{2}\), including \(\frac{1}{2}\), and goes towards infinity. In contrast, \(1 - 2x \geq 0\) translates to \(( -\infty, \frac{1}{2}]\), indicating \(x\) value lies from negative infinity up to \(\frac{1}{2}\), including \(\frac{1}{2}\).

Interval notation makes it clear and quick to see the number line representation of an inequality solution.

Linear Equations

A linear equation is one of the most straightforward types of algebraic equations. It involves variables raised only to the first power and typically has the form \(ax + b = 0\), where \(a\) and \(b\) are constants.

Linear equations represent a straight line when graphed on a coordinate plane. This is because they have a constant rate of change or slope. Solving linear equations is all about isolating the variable, usually \(x\), on one side of the equation to find the specific value it must take to make the equation true.

Take the equation \(1 - 2x = 0\). To find \(x\), we rearrange the equation:

• Add \(2x\) to both sides: \(1 = 2x\)
• Divide by 2: \(x = \frac{1}{2}\)

Once solved, this shows there is a single value for \(x\) that satisfies the equation. Understanding how to manipulate these kinds of expressions is fundamental to tackling more complex algebraic concepts.

Inequalities

Inequalities are mathematical expressions to compare values and show the relationship between them. Unlike equations, which assert two expressions are equal, inequalities tell us about the less than, greater than, less than or equal to, or greater than or equal to relations.

The four basic types of inequalities are:

• \(<\)
• \(>\)
• \(\leq\)
• \(\geq\)

When solving inequalities like \(1 - 2x \leq 0\) or \(1 - 2x \geq 0\), the goal is to isolate \(x\) just as you would in an equation. The approach:

• Start by adding or subtracting terms to get the inequality into a simpler form.
• Divide or multiply by a constant to solve for the unknown while keeping in mind that multiplying or dividing by a negative number flips the inequality sign.

With \(1 - 2x \leq 0\) and \(1 - 2x \geq 0\), applying these steps reveals the range of values for \(x\) that satisfy these inequalities.

Understanding inequalities and their graphical representation on the number line helps immensely in mastering algebra and forms the foundation for solving more dynamic mathematical problems.