Problem 40 - 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 \(x \geq 5\). \(6 x-4 \leq 5 x-4\)

Step-by-Step Solution

Answer

The inequality is \(x \leq 0\).

Step 1: Simplify the Inequality

Start by simplifying the inequality given in the problem. The inequality is \(6x - 4 \leq 5x - 4\). Subtract \(5x\) from both sides to isolate the variable \(x\) on one side of the inequality.\[6x - 5x - 4 \leq -4\]

Step 2: Solve for the Variable

Now simplify the left side of the inequality. Subtract \(5x\) from \(6x\) to get \(x\).\[x - 4 \leq -4\]Next, add \(4\) to both sides to solve for \(x\).\[x - 4 + 4 \leq -4 + 4\]This simplifies to:\[x \leq 0\]

Step 3: Express Interval as an Inequality

The simplified inequality is \(x \leq 0\). This can be interpreted as the interval \((-\infty, 0]\), where \(x\) takes on all values less than or equal to zero.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It uses intervals to convey the set of numbers that satisfy a particular condition. Instead of using words or phrase-based descriptions, interval notation offers a concise mathematical expression.

When you see an interval, it usually comes in two parts - the lower and the upper bounds. These bounds are written inside brackets:

• Square brackets "[ ]" denote that the endpoint is included, known as a "closed interval."
• Round brackets "( )" mean the endpoint is not included, referred to as an "open interval."

For example, the interval \([a, b)\) means all numbers "x" between "a" and "b", including "a" but not including "b".

In the case of our example, the solution \(x \leq 0\) translates to the interval \((-\infty, 0]\). This tells us that "x" can be any real number less than or equal to zero, extending infinitely to the left.

Solving Inequalities

Solving inequalities is like solving regular equations, but with extra attention to the inequality sign.

There are a few key rules to keep in mind:

• Treat the inequality sign (>, >=, <, <=) like an equal sign while adding or subtracting terms from both sides.
• If you multiply or divide by a positive number, the inequality sign remains the same.
• However, if you multiply or divide by a negative number, you must "flip" the inequality sign.

Let's say you have the inequality \(6x - 4 \leq 5x - 4\).

• Start by isolating terms involving "x" on one side.
• Simplify what you can, making sure to keep the inequality sign in mind.

In our example, by subtracting \(5x\) from both sides, it simplified the equation down, leading us to easily find \(x \leq 0\).

So, pay close attention to signs and operations, and follow these rules to solve inequalities correctly.

Variable Isolation

Variable isolation is the process of manipulating an equation or inequality to have the variable of interest on one side. This allows you to clearly identify what range of values satisfy the given mathematical condition. Think of it as "freeing" the variable from other numbers or terms.

In order to isolate a variable, perform a series of steps:

• Add or subtract terms to both sides to start grouping like terms together.
• Use multiplication or division to eliminate coefficients beside the variable. Remember, changing the sign (perhaps due to multiplying/dividing by a negative) can affect your inequality direction.

For the inequality \(6x - 4 \leq 5x - 4\), we simplify by removing \(5x\) from both sides, resulting in \(x - 4 \leq -4\).
The next step involves getting "x" by itself, which here means adding 4 to both sides, finally yielding \(x \leq 0\).
This simple system of step-by-step operations ensures variables can be effectively isolated for clarity in expressions and inequalities.