Problem 17 - Algebra for College Students

Question

Solve each of the inequalities and express the solution sets in interval notation. \(x \geq 3.4+0.15 x\)

Step-by-Step Solution

Answer

The solution is \([4, \infty)\).

Step 1: Rearrange the Inequality

Start by subtracting \(0.15x\) from both sides to get the terms involving \(x\) on one side of the inequality.\[x - 0.15x \geq 3.4\]This simplifies to:\[0.85x \geq 3.4\]

Step 2: Solve for x

Divide both sides of the inequality by \(0.85\) to solve for \(x\).\[x \geq \frac{3.4}{0.85}\]Calculate the right side of the inequality to find:\[x \geq 4\]

Step 3: Express the Solution in Interval Notation

The solution \(x \geq 4\) indicates that \(x\) is all values greater than or equal to 4. In interval notation, this is expressed as:\[[4, \infty)\]

Key Concepts

Interval Notation

Interval notation is a compact way of expressing a range of values that an inequality represents. This becomes particularly handy when the solution involves a continuous set of numbers rather than individual points. For instance, if you have a solution like \( x \geq 4 \), it means \( x \) can take any value starting from 4 up to infinity.
This range is denoted in interval notation as \([4, \infty)\).

• The bracket \([\) indicates that 4 is included in the solution.
• The parenthesis \()\) after infinity shows that there is no upper bound since infinity is not a number you can reach.

Using interval notation simplifies inequalities because it provides a clear picture of the values involved without listing each one. It's concise and eliminates the need for lengthy descriptions of number ranges.

Solving Inequalities

Solving inequalities is a critical math skill that helps you determine the range of possible solutions for a given condition. An inequality expresses a relationship where one value is larger or smaller than another. In the example \( x \geq 3.4 + 0.15x \), you aim to find all values of \( x \) that satisfy this condition.
The solution process involves similar steps as solving equations, but with extra attention to the inequality sign. For instance:

• Rearrange all terms so variables are on one side and constants on the other.
• Perform operations such as addition, subtraction, multiplication, or division, but remember: multiplying or dividing by a negative number reverses the inequality.

Once solved, the inequality will give you a range of values for \( x \) that make the inequality true.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or inequalities. This skill is essential in converting complex statements into manageable forms. Let's break it down using the inequality \( x \geq 3.4 + 0.15x \):

• First, isolate the variable. This means moving all terms involving \( x \) to one side. Subtract \( 0.15x \) from both sides to align terms. This results in \( 0.85x \geq 3.4 \).
• Next, divide both sides by \( 0.85 \) to solve for \( x \). So, you have \( x \geq 4 \).

These steps are an example of algebraic manipulation where the main goal is to simplify and solve expressions. By systematically applying operations, you can make sense of, and resolve, seemingly complex mathematical situations.