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Chapter 3: Problem 70

Solve each inequality. $$ -5 \leq-4 x-1+4 x $$

Short Answer

Expert verified
The inequality is always true.

Step by step solution

01

Simplify the Expression

Start by simplifying the expression on the right side of the inequality. It is given as \(-4x - 1 + 4x\). The \(-4x\) and \(+4x\) cancel each other out, simplifying to \(-1\). Thus, the inequality becomes \(-5 \leq -1\).
02

Evaluate the Inequality

Now compare the simplified inequality \(-5 \leq -1\). Since \(-5\) is indeed less than or equal to \(-1\), the inequality holds true.

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Key Concepts

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

Simplifying Expressions
When working with inequalities in elementary algebra, one of the first steps is often to simplify expressions. Simplifying an expression helps you to clearly see what's being asked in the problem and to manipulate it more easily. In the given exercise, we have the expression
  • \(-4x - 1 + 4x\)
The task is to simplify this expression. Here's how you do it:
  • Identify like terms: In this expression, the terms \(-4x\) and \(+4x\) are like terms.
  • Combine these terms: Since they are opposites, they cancel each other out.
This leaves you with the simplified expression
  • \(-1\)
By simplifying, you make further steps in solving the inequality much simpler.
Mathematical Inequality
A mathematical inequality is a statement that compares two values or expressions, showing one is greater than, less than, equal to, or not equal to the other. In algebra, inequalities are ubiquitous and essential for solving many real-world problems.In our exercise, after simplifying the expression, we end up with the inequality:
  • \(-5 \leq -1\)
The question is whether this statement is true. Inequality symbols include:
  • \(<\): Less than
  • \(>\): Greater than
  • \(\leq\): Less than or equal to
  • \(\geq\): Greater than or equal to
Here, \(-5\) is indeed less than or equal to \(-1\), so our statement holds true. Understanding inequalities helps in determining ranges or feasible solutions when multiple conditions are involved.
Elementary Algebra
Elementary algebra forms the foundation of understanding math problems such as our inequality exercise. It allows you to manipulate numbers and variables to find unknowns and make comparisons.Key components of elementary algebra include:
  • Understanding variables: Symbols like \(x\) represent unknown numbers.
  • Performing basic operations: Addition, subtraction, multiplication, and division on expressions.
  • Simplifying expressions: Combining like terms to reduce expressions to simpler forms.
  • Solving equations or inequalities: Finding values of variables that satisfy given conditions.
With these skills, solving inequalities becomes straightforward. In our example, we combined like terms and verified our statement. By understanding these basic principles, algebra equations and inequalities become much easier to tackle effectively.

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