Problem 78 - Intermediate Algebra

Question

Let \(s(x)=\frac{1}{4} x-\frac{1}{2}\) and \(g(x)=\frac{1}{2} x-\frac{2}{3} .\) Find all values of \(x\) for which \(s(x) \geq g(x)\).

Step-by-Step Solution

Answer

The solution is \(x \leq \frac{2}{3}\).

Step 1: Set Up the Inequality

We begin by setting up the inequality from the functions provided: \(s(x) \geq g(x)\). This gives us: \( \frac{1}{4}x - \frac{1}{2} \geq \frac{1}{2}x - \frac{2}{3} \).

Step 2: Move Terms Involving x to One Side

Subtract \(\frac{1}{2}x\) from both sides to move all terms involving \(x\) to one side of the inequality: \( \frac{1}{4}x - \frac{1}{2}x \geq -\frac{2}{3} + \frac{1}{2} \).

Step 3: Simplify the Terms Involving x

Calculate \(\frac{1}{4}x - \frac{1}{2}x\) which is \(-\frac{1}{4}x\). The inequality is now: \(-\frac{1}{4}x \geq -\frac{2}{3} + \frac{1}{2} \).

Step 4: Simplify the Constant Terms

Calculate \(-\frac{2}{3} + \frac{1}{2}\). First, find common denominators: \(-\frac{2}{3} = -\frac{4}{6}\) and \(+\frac{1}{2} = +\frac{3}{6}\). Adding them yields \(-\frac{1}{6}\). The inequality becomes \(-\frac{1}{4}x \geq -\frac{1}{6}\).

Step 5: Solve for x by Inverting the Inequality

Divide both sides by \(-\frac{1}{4}\) to solve for \(x\). Remember that dividing by a negative flips the inequality sign: \(x \leq \frac{1}{6} \div \frac{1}{4}\). Simplifying gives \(x \leq \frac{4}{6}\), which simplifies to \(x \leq \frac{2}{3}\).

Key Concepts

Linear Inequalities

Linear inequalities involve mathematical expressions that use inequality symbols (<, >, ≤, or ≥) to compare linear expressions. They tell us about the relationship between expressions in terms of their magnitude. Generally, the key steps to solve linear inequalities are similar to those for linear equations, with the added complexity of dealing with the inequality sign.

• Set Up the Inequality: Just like with equations, you need to write down the inequality first. For the example exercise, we set up the inequality as \( \frac{1}{4}x - \frac{1}{2} \geq \frac{1}{2}x - \frac{2}{3} \).
• Isolate Variable Terms: Move all terms involving the variable to one side. For instance, subtract \( \frac{1}{2}x \) from both sides in our exercise. The key here is to ensure that you're simplifying correctly and keeping the inequality in balance.
• Simplify: Combine like terms and simplify the inequality. In this case we deal with fractions such as simplifying \( \frac{1}{4}x - \frac{1}{2}x \).
• Solving: If you need to divide or multiply the inequality by a negative number, remember to flip the inequality sign. This is critical for linear inequalities, as seen when dividing through by \(-\frac{1}{4}\), which changes \( \geq \) to \( \leq \).
• Check Your Solution: Finally, it is always wise to check your solution, potentially by choosing a test value within your solution set and substituting it back into the original inequality to verify.

Understanding these steps not only helps in solving one inequality but also builds a strong foundation for more complex inequalities.

Linear Equations

Linear equations are equations where the highest power of the variable is one. They form a straight line when graphed and are foundational to algebra because they represent steady relationships.

• Understanding the Form: Linear equations typically look like \( ax + b = 0 \), where \(a\) and \(b\) are constants. They can be manipulated using operations such as addition, subtraction, multiplication, and division.
• Slope-Intercept Form: Many linear equations are written in the form \( y = mx + c \), where \(m\) represents the slope and \(c\) the y-intercept. This form is useful for graphing and understanding the relationship between variables.
• Solution Methods: We solve linear equations by isolating the variable on one side of the equation. This often involves shifting terms from one side to the other and simplifying. Tools like cross-multiplication can also apply in specific cases, especially with equations involving fractions.

While linear equations are simpler than inequalities, mastering them is crucial as they form the building block for understanding and solving more complex problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are the language of algebra and provide a way to represent general relationships mathematically.

• Components: An algebraic expression can include constants (like numbers), variables (like \(x\) and \(y\)), coefficients (numbers multiplying variables), and operations (such as addition or subtraction).
• Examples: In this exercise, the algebraic expressions \( \frac{1}{4}x - \frac{1}{2} \) and \( \frac{1}{2}x - \frac{2}{3} \) involve linear terms and constants. Each of these represents a unique line when plotted on a graph.
• Simplification: Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form. For instance, combining terms such as \( \frac{1}{4}x - \frac{1}{2}x \) requires finding a common factor.
• Use in Problems: Expressions are not equations or inequalities until they involve an equality or inequality sign. They're used to form and manipulate the problems that allow us to solve for unknowns.

Understanding how to work with algebraic expressions is vital for solving equations and inequalities, as they form the basic element of these mathematical statements.