Problem 137 - College Algebra

Question

This will help you prepare for the material covered in the next section. Is \(-1\) a solution of \(3-2 x \leq 11 ?\)

Step-by-Step Solution

Answer

Yes, \(-1\) is a solution of the given inequality.

Step 1: Substitute the value

First, substitute the number -1 for the variable x. The inequality thus becomes \(3 - 2(-1) \leq 11\). Then simplify it.

Step 2: Simplify

When you simplify \(3 - 2(-1)\), you get \(3 - (-2)\), which simplifies to \(3+2=5\). That gives us the simplified inequality \(5 \leq 11\).

Step 3: Verify

5 is indeed less than or equal to 11, and hence \(-1\) is a solution of the inequality.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and mathematical operations. In the context of inequalities, they serve as the building blocks that express relationships between quantities. Understanding the role each component plays in an expression is key to solving inequalities effectively.
For example, in the inequality given in the exercise, \( 3 - 2x \), is an algebraic expression. Here:

• "3" is a constant term.
• "-2x" is a variable term, consisting of the coefficient "-2" and the variable "x".

When you substitute a value for "x," like -1 in this problem, you are evaluating the expression numerically.
This substitution helps you transform the abstract inequality into a statement you can easily verify. In our inequality, substituting \(-1\) for "x" simplifies the expression, making it possible to understand the relationship between its components.

Finding the Solution of an Inequality

An inequality is much like an equation, but instead of an equal sign, it uses inequality symbols (\(<\), \(>\), \(\leq\), \(\geq\)) to compare expressions. Solving an inequality involves finding all values that fulfill the inequality condition.
In our problem, the inequality is \( 3 - 2x \leq 11 \). This means we need to find the values of "x" that make the inequality true. By substituting the number \(-1\) for "x", the inequality transforms into \( 3 - 2(-1) \leq 11 \).
Through substitution, you ended up with \( 5 \leq 11 \), which is a true statement. Therefore, \(-1\) is indeed a solution of the inequality because it satisfies the original inequality condition.
Solving inequalities often requires checking if substituting a given value for the variable produces a true statement. This approach verifies if a specific number belongs to the set of solutions for that inequality.

The Simplification Process

The simplification process is crucial in solving inequalities as it makes the expressions manageable and easier to comprehend. When simplifying an algebraic expression, your goal is to combine like terms and perform operations to rewrite the expression in a simpler form.
In our example, the simplification started after substituting -1 for "x" in \( 3 - 2x \), resulting in \( 3 - 2(-1) \).
Here are the steps for simplification:

• Multiply the coefficient \(-2\) by the substituted value \(-1\) to get \(-2(-1) = 2\).
• Subtract \(-2\) from \(3\), realizing that subtracting a negative is the same as adding a positive, thus \(3 - (-2) = 3 + 2\).
• Combine the constants to simplify further, yielding \(5\).

This simplification allowed us to see that \( 5 \leq 11 \) is indeed true, confirming that the original inequality holds. Understanding the steps of simplification helps ensure that you accurately derive solutions from more complex expressions.