Problem 80 - Intermediate Algebra

Question

Let \(y_{1}=0.8 x-1.1\) and \(y_{2}=3.1-0.6 x .\) Find all values of \(x\) for which \(y_{1}\) does not exceed \(y_{2}\).

Step-by-Step Solution

Answer

The values of \(x\) such that \(y_1\) does not exceed \(y_2\) are \(x \leq 3\).

Step 1: Set up the inequality

We want to find all values of \(x\) for which \(y_1 \leq y_2\). Start by writing the inequality: \(0.8x - 1.1 \leq 3.1 - 0.6x\).

Step 2: Isolate the variable terms

Add \(0.6x\) to both sides of the inequality to collect the \(x\) terms on one side: \(0.8x + 0.6x - 1.1 \leq 3.1\). This simplifies to \(1.4x - 1.1 \leq 3.1\).

Step 3: Isolate the constant terms

Add 1.1 to both sides to move the constant from the left side to the right: \(1.4x \leq 3.1 + 1.1\), which simplifies to \(1.4x \leq 4.2\).

Step 4: Solve for x

Divide both sides by 1.4 to solve for \(x\): \(x \leq \frac{4.2}{1.4}\). Calculate the division to find \(x \leq 3\).

Step 5: Interpret the result

The inequality \(x \leq 3\) means that the values of \(x\) for which \(y_1\) does not exceed \(y_2\) are all values less than or equal to 3.

Key Concepts

Solving Inequalities

When solving inequalities, it's important to approach them similarly to equations. The goal is to find the range of values a variable can take. Let's start with understanding basic operations and rules involved in solving inequalities.

In this exercise, we were given two linear expressions: \( y_1 = 0.8x - 1.1 \) and \( y_2 = 3.1 - 0.6x \). We needed to solve the inequality for \( y_1 \leq y_2 \). This requires us to manipulate the inequality until we isolate \( x \) on one side. Unlike equations, however, inequalities have special properties:

• If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
• Adding or subtracting the same number from both sides does not affect the direction of the sign.

To solve \( 0.8x - 1.1 \leq 3.1 - 0.6x \), we moved all \( x \) terms to one side and constants to the other by adding \( 0.6x \) and \( 1.1 \) to both sides, resulting in \( 1.4x \leq 4.2 \). Finally, we divided by \( 1.4 \) to find \( x \leq 3 \). This result shows the solution set on a number line.

Linear Equations

Linear equations are any equation that makes a straight line when graphed. They are incredibly foundational in algebra.

The equations \( y_1 = 0.8x - 1.1 \) and \( y_2 = 3.1 - 0.6x \) are examples of linear equations. Each term involves only a constant or a single variable multiplied by a constant. The graph of these equations would each form a straight line.

• The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• For \( y_1 \), the slope is \( 0.8 \) and the y-intercept is \( -1.1 \).
• For \( y_2 \), the slope is \( -0.6 \) and the y-intercept is \( 3.1 \).

Understanding these parameters helps when analyzing how each line behaves, especially in determining when one line is above or below another, which is crucial when solving inequalities involving linear equations.

Mathematical Reasoning

Mathematical reasoning plays a key role in solving expressions and inequalities. It involves logical thinking and making connections between concepts to find solutions.

In our exercise, we applied mathematical reasoning by understanding how to handle inequality transformation. This involves using properties of inequalities and operations, while keeping the main objective to make \( x \) the subject.

• We recognized the need to bring terms involving \( x \) to one side. This step is crucial because it simplifies the inequality to a simpler form: \( 1.4x \leq 4.2 \).
• Then, we intuitively divided by a positive constant (1.4), which doesn't involve changing the inequality sign, to solve for \( x \).

Understanding each of these logical steps not only ensures that the process is right but also enhances our ability to apply similar reasoning to more complex problems in mathematics.