Problem 39 - Algebra for College Students

Question

Express each interval as an inequalit using the variable \(x\). For example, we can express the inter val \([5, \infty)\) as \(x \geq 5\). \(5 x+2 \geq 4 x+6\)

Step-by-Step Solution

Answer

x ≥ 4, interval is [4, ∞).

Step 1: Simplify the Inequality

First, let's simplify the inequality given: \(5x + 2 \geq 4x + 6\). Begin by eliminating \(x\) from one side. Subtract \(4x\) from both sides of the inequality to simplify:\[ 5x + 2 - 4x \geq 4x + 6 - 4x \] This simplifies to:\[ x + 2 \geq 6 \]

Step 2: Solve for x

Now, isolate \(x\) by subtracting \(2\) from both sides:\[ x + 2 - 2 \geq 6 - 2 \]Thus, we find:\[ x \geq 4 \]

Step 3: Express the Interval

The inequality \(x \geq 4\) represents the interval \([4, \infty)\) in interval notation. This means that \(x\) includes all values starting from \(4\) and goes to infinity.

Key Concepts

Solving Inequalities

Understanding how to solve inequalities is an important skill in mathematics. An inequality states that one side is greater or lesser than the other and can be represented using symbols like ">", "<", "≥", and "≤". The process of solving these inequalities involves rearranging or simplifying terms to find the range of values that satisfy the inequality.

To solve an inequality like the example given: \(5x + 2 \geq 4x + 6\), you need to start by simplifying. This means getting all the terms involving the variable \(x\) on one side of the inequality and all the constant terms on the other side. The goal is to isolate \(x\) to determine its possible values.

One important thing to remember is that operations you perform when solving inequalities (like adding, subtracting, multiplying, or dividing) follow the same rules as solving equations, with one key exception. If you ever multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This needs to be kept in mind to avoid mistakes.

Interval Notation

Interval notation is a shorthand way to describe a set of numbers along a number line. It's often used to express the solution to an inequality. This notation uses brackets to show whether endpoints are included (or not) in a set. For example, the interval \([4, \infty)\) represents all numbers starting from 4 and extending to infinity including 4 itself.

The two kinds of brackets used are:

• Square brackets \([ ]\) indicates that the endpoint value is included in the interval, often called "closed."
• Parentheses \(( )\) indicates that the endpoint is not included, often referred to as "open."

When using "\(\infty\)" or "\(-\infty\)", we always use a parenthesis because infinity is not a point you can actually reach or include. Interval notation is a concise and clear method to communicate ranges of solutions.

Simplifying Expressions

Simplifying expressions is a key technique in solving any mathematical problem, especially inequalities. Simplifying means to make the expression as straightforward as possible by eliminating unnecessary terms and combining like items. This allows you to focus on the main components of the inequality.

For example, in the expression \(5x + 2 \geq 4x + 6\), to simplify, you combine like terms and then rearrange to isolate \(x\). By subtracting \(4x\) from both sides, you arrive at \(x + 2 \geq 6\). From here, subtract 2 from both sides to solve for \(x\), giving the simple form \(x \geq 4\).

Simplifying is crucial because it reduces the clutter in the inequality, making it easier to understand and solve. It's a step where attention to detail pays off, ensuring that the solution reflects all the correct values.