Problem 29 - 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 \(x \geq 5\). \(2+6 x>-10\)

Step-by-Step Solution

Answer

The inequality is \(x > -2\).

Step 1: Simplify the Inequality

We start with the inequality given: \(2 + 6x > -10\). To isolate the term with \(x\), subtract 2 from both sides: \(6x > -10 - 2\). This simplifies to \(6x > -12\).

Step 2: Solve for x

Now, divide each side of the inequality by 6 to solve for \(x\): \(x > \frac{-12}{6}\). Simplifying the right side, you get \(x > -2\).

Step 3: Write the Inequality

The inequality \(x > -2\) represents the original interval as an inequality. This means that \(x\) is any number greater than \(-2\).

Key Concepts

Interval Notation

Interval notation is a way of representing a range of numbers on the number line. It uses brackets and parentheses to indicate the endpoints of an interval. For example, the interval \([-2, \infty)\) means all numbers greater than or equal to -2 and extending infinitely upwards.
There are two main types of brackets used:

• Round Parentheses: \((a, b)\) - This means the interval does not include the endpoints, acting more like an open interval. For example, \((-2, 3)\) includes all numbers greater than -2 and less than 3, but not -2 or 3 themselves.
• Square Brackets: \([a, b]\) - This indicates that the interval includes the endpoints. So, \([-2, 3]\) is the interval from -2 to 3, including both -2 and 3.

When an interval extends to infinity, such as \([5, \infty)\), a round parenthesis is always used with \( \infty \) since infinity is a concept and not an actual number, thus cannot be included in an interval.

Solving Inequalities

Solving inequalities is similar to solving equations but has some unique properties. An inequality tells us about the relative size of two values. For our example, we have the inequality \(2 + 6x > -10\). The goal is to find all the values of \(x\) that make this inequality true.
Here's how to solve it:

• First, simplify the inequality to isolate the variable term. For instance, you start by removing constants from one side, like subtracting 2 from both sides in \(2 + 6x > -10\), which gives \(6x > -12\).
• Next, solve for the variable just like you would in a regular equality. Divide both sides by the coefficient of \(x\), ensuring you maintain the inequality's direction. This step turns \(6x > -12\) into \(x > -2\).
• Remember, if you ever multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

This process helps you find all possible solutions that satisfy the inequality, giving a problem its true context within a range of values.

Variable Isolation

Variable isolation is the process of rearranging an equation or inequality so that the variable stands alone on one side. This technique is crucial in solving mathematical problems, as it allows you to find the specific values of variables.
For example, in \(2 + 6x > -10\), we isolate \(x\) by first removing any constants or coefficients attached. You'd subtract 2 from both sides to get \(6x > -12\). This step helps you focus only on the part of the inequality involving \(x\).
Then, to further isolate \(x\), divide both sides by 6, simplifying to \(x > -2\). This leaves \(x\) by itself on one side, clearly showing which values of \(x\) satisfy the inequality.
Key points to remember:

• Keep operations balanced: Whatever you do to one side of the inequality, do to the other.
• If you multiply or divide by a negative number, remember to flip the inequality sign.
• Simplify step by step to avoid mistakes, ensuring you isolate the variable correctly.

Mastering variable isolation makes solving both equations and inequalities more systematic and intuitive.