Problem 123 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(-4 x<-20\) is equivalent to \(x>-5\).

Step-by-Step Solution

Answer

The statement is false. The correct inequality is \(x > 5\).

Step 1: Identify the Inequality

The given inequality is \(-4x < -20\).

Step 2: Solve the Inequality

We solve the inequality by isolating 'x'. To do this, we need to divide both sides of the inequality by -4. However, we must remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. So, \(-4x/-4 > -20/-4\). This simplifies to \(x > 5\).

Step 3: Compare the Result to the Given Statement

The given statement states that the inequality is equivalent to \(x > -5\). Our solution shows that this is not correct. The true statement is \(x > 5\).

Key Concepts

Solving Inequalities

Inequalities in algebra are like little puzzles waiting to be solved. The aim is to find the range of values that can satisfy the expression. Imagine the inequality \(-4x < -20\). We start by tackling what’s "trapping" the variable.
Since our goal is to have just \(x\) on one side, we need to deal with the \(-4\) that's multiplied with it. A handy way to free \(x\) is to divide both sides by \(-4\). This way, the inequality becomes simpler and you uncover the possible range for the variable.
So, solving inequalities is all about making the variable the star of the show. Just remember that each action on one side needs to be mirrored on the other!

Negative Number Rule in Inequalities

A cool fact about inequalities is what happens when you introduce negative numbers. When you multiply or divide by a positive number, nothing much alters, apart from the numbers themselves. However, when you multiply or divide an inequality by a negative number, the inequality sign flips. Sounds intriguing? It truly is!
Take our inequality \(-4x < -20\). Here, dividing both sides by \(-4\), the sign changes from '<' to '>'. This twist is crucial, and forgetting to do so can lead to wrong answers.

• Whenever you multiply or divide by a negative: flip the sign!
• Keep the variable isolated to read off the range of solutions easily.

Equivalent Statements in Inequalities

Equivalent statements mean that, despite potentially looking different, the statements represent the same set of solutions. In inequalities, though, a slip in equivalence can lead to false claims.
Consider the inequality we solved earlier: \(x > 5\). This tells us any number greater than 5 satisfies the original inequality \(-4x < -20\).
If a statement says the inequality is equivalent to \(x > -5\), it misleads as the range of numbers greater than -5 includes numbers that do not satisfy the original condition.
Checking equivalence is like verifying a magic trick; you need to make sure everything is spot on, or you might end up with a statement that looks fine but gets the magic totally wrong!