Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ -2-3 x \geq 2 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{4}{3}]\).

Step 1: Move Constant to Right Side

Start by isolating the term with the variable on one side of the inequality. Add 2 to both sides to eliminate the constant term from the left side: \[-2 - 3x + 2 \geq 2 + 2\]which simplifies to:\[-3x \geq 4\]

Step 2: Solve for x by Dividing

Divide both sides of the inequality by -3. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign:\[-3x \geq 4\] becomes \[x \leq -\frac{4}{3}\].

Step 3: Express the Solution as an Interval

Use interval notation to express the solution. Since \(x\) is less than or equal to \(-\frac{4}{3}\), the interval is:\[ (-\infty, -\frac{4}{3}] \].

Key Concepts

Interval Notation

Understanding how to express solutions in interval notation is crucial when solving inequalities. Interval notation is a way of writing subsets of the real number line. It's a concise way to denote ranges of numbers. For the inequality where you have a variable on one side, like in our exercise, and numbers on the other side, you use brackets and parentheses.

• Parentheses, () are used when the endpoint is not included in the interval, meaning the inequality is "less than" or "greater than".
• Brackets, [] are used when the endpoint is included in the interval, signifying either "less than or equal to" or "greater than or equal to".

In the given exercise, the solution to the inequality was expressed as \[ (-\infty, -\frac{4}{3}] \]. This interval tells us that all numbers less than or equal to \(-\frac{4}{3} \) are solutions to the inequality, extending negatively to infinity. The use of \(-\infty \) with a parenthesis means it's not an actual number; rather, it simply indicates that the interval continues infinitely.

Inequality Reversal

An essential rule when solving inequalities is remembering to reverse the inequality sign when multiplying or dividing both sides by a negative number. This concept can lead to significant misunderstandings if overlooked. Here's why this rule exists:
Suppose you have an inequality statement like \(-3x \geq 4\). You must divide by \(-3\) to solve for \(x\). When you execute this operation, you reverse the {'\geq'} to {'\leq'} because multiplying or dividing by a negative affects the order. This is because the negative sign reflects numbers across zero, flipping their direction on the number line.
Imagine comparing two numbers: \(2\) and \(5\). If you multiply both by \(-1\), you get \(-2\) and \(-5\). Originally, \(2\) was less than \(5\), but now \(-2\) is greater than \(-5\). Therefore, always remember to switch signs when dealing with negative multiplication or division.

Dividing by Negative Numbers

Dividing by a negative number in inequalities reshapes the entire landscape of possible solutions. The act of reversing the inequality when performing this operation is a unique rule for inequalities, distinct from equations.
When we divided \(-3x \geq 4\) by \(-3\), the inequality symbol \(\geq\) was flipped to \(\leq\) at the step where the solution emerged as \(x \leq -\frac{4}{3}\). This flip is what differentiates inequalities from regular equations, which lack such considerations.
Understanding the Flip:

• Negative multiplication or division changes the sense of the inequality.
• It impacts the direction of the relationship between expressions.

Make sure to apply this rule every time you encounter a situation that requires multiplying or dividing both sides of an inequality by a negative number.