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Chapter 3: Problem 16

Solve each inequality. $$-w>-6$$

Short Answer

Expert verified
w < 6

Step by step solution

01

Understand the inequality

The inequality given is -w > -6. The objective is to isolate the variable w on one side of the inequality.
02

Multiply both sides by -1

To isolate w, multiply both sides of the inequality by -1. Remember that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign reverses. -1(-w) < -1(-6) This simplifies to: w < 6

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In equations and inequalities, these symbols typically represent variables. In the given problem, the variable is represented by the letter \(w\). The fundamental objective of algebra is to solve for these variables by isolating them on one side of the equation or inequality. This often involves operations such as addition, subtraction, multiplication, and division. By following a structured approach, we can simplify complex expressions into more understandable terms.
Inequality Reversal
In the realm of inequalities, an important concept to grasp is the reversal of the inequality sign. This reversal occurs when you multiply or divide both sides of an inequality by a negative number. Let's break this down with the given inequality:
  • The original inequality is \(-w > -6\).
  • To isolate \(w\), we need to get rid of the negative sign. We do this by multiplying both sides by -1.
  • When we multiply both sides by -1, it is necessary to reverse the direction of the inequality symbol.
  • So, \(-1 * -w\) becomes \(w\) and \(-1 * -6\) becomes 6, changing the inequality to \(w < 6\).
This rule is crucial to remember, as failing to reverse the inequality sign can lead to incorrect solutions.
Variable Isolation
Isolating the variable is a core objective when solving equations and inequalities. Typically, this involves performing operations that move all instances of the variable to one side of the inequality or equation. Here's how it works in our problem:
  • We start with the inequality \(-w > -6\).
  • Our goal is to get \(w\) by itself on one side of the inequality.
  • To do this, we multiply both sides by -1, while remembering to reverse the inequality sign.
  • This transforms \(-w > -6\) into \(w < 6\).
Now, the variable \(w\) is isolated, and we can clearly see that \(w\) must be less than 6. Isolating the variable makes it much easier to understand the solutions to inequalities and equations.

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Most popular questions from this chapter

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. The width of a rectangle is 10 less than five times the length. If the perimeter of the rectangle is 100 yd, find the dimensions of the rectangle.

Translate the given phrase or sentence algebraically. \(x\) is at least 5

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Explain in words what the inequality \(2
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