Problem 34 - 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 \(x \geq 5\). \(12<2-5 x\)

Step-by-Step Solution

Answer

The inequality solution is \(x < -2\).

Step 1: Isolate the variable term

Start by moving the constant from the right-hand side to the left-hand side of the inequality. To do this, subtract 2 from both sides of the inequality:\[12 - 2 < 2 - 5x - 2\] This simplifies to:\[10 < -5x\]

Step 2: Solve for the variable

Now, to isolate \(x\), divide both sides of the inequality by -5. Remember, when you divide or multiply an inequality by a negative number, you must reverse the inequality sign:\[\frac{10}{-5} > x\] This simplifies to:\[-2 > x\] or \[x < -2\]

Step 3: Interpret the inequality

The inequality \(x < -2\) describes the solution set in terms of \(x\). This tells us that the variable \(x\) can take any real number less than -2.

Key Concepts

Interval Notation

Intervals are a way of describing a range of numbers in mathematics. Instead of listing every possible value, interval notation gives us a concise way of showing which numbers are included in a set.

In interval notation, we use parentheses

• Parentheses \(( )\) indicate that an endpoint is not included, called "open."
• Brackets \([ ]\) indicate that an endpoint is included, called "closed."

For example, \((−\infty, 5]\) includes all numbers less than or equal to 5 but does not include negative infinity (as it's a concept rather than a number). In inequalities, an interval like \([a, b]\) corresponds to an inequality like \(a \leq x \leq b\). Understanding how to move between interval notation and inequalities is a useful skill when solving math problems.

Solving Inequalities

Solving inequalities is similar to solving regular equations. The goal is to isolate the variable, providing a range of solutions rather than just one. Here's the approach:

• Start by moving terms to create a simpler expression.
• Simplify by combining like terms.
• Use addition, subtraction, multiplication, or division to isolate the variable.

Always remember that the operations you perform on both sides of the inequality keep it balanced. The result will need interpretation to understand which numbers satisfy the condition given by the inequality.

Inequality Sign Reversal

One key aspect of solving inequalities is the rule about reversing the inequality sign. Whenever we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality sign.

This is crucial for achieving the correct solution. For example, if we have \(-5x > 10\), dividing both sides by \(-5\) requires reversing the '>' sign to '<', resulting in \(x < -2\).

Failing to reverse the sign would lead to an incorrect answer, representing a small but very important part of solving inequalities correctly.

Real Numbers

The set of real numbers includes all rational and irrational numbers, providing a complete picture of continuous values. Real numbers can be represented on a number line and include:

• Integers such as -2, 0, 1
• Fractions or decimals, such as 1/2 or 3.14
• Non-repeating, non-terminating decimals such as \(\pi\) or \(\sqrt{2}\)

When solving inequalities involving real numbers, each solution represents not just one value, but a whole range of possibilities. For example, the solution \(x < -2\) indicates that any real number less than -2 satisfies the inequality. This concept is vital in understanding the behavior of number sets in mathematical contexts.