Problem 24 - Algebra Form and Function

Question

For what values of \(x\) are the quantities negative? $$ x-7 $$

Step-by-Step Solution

Answer

The expression \(x-7\) is negative for values of \(x\) that are less than 7, or \(x < 7\).

Step 1: Set up the inequality

Write the inequality with the given expression, \(x-7\): $$ x-7 < 0 $$

Step 2: Solve the inequality for \(x\)

Add \(7\) to both sides of the inequality: $$ x-7+7 < 0+7 $$ Simplifying, we get: $$ x < 7 $$

Step 3: Write the answer

The values of \(x\) for which the expression \(x-7\) is negative are given by: $$ x < 7 $$

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions involving symbols like \(<, >, \leq, \) and \(\geq\). They show relationships where one quantity is smaller or larger than another. In the context of solving inequalities, the goal is to determine the values that satisfy these conditions. For example, if we have \(x - 7 < 0\), it implies that \(x - 7\) is less than zero, meaning it's negative. To solve inequalities, we often manipulate them similarly to equations. The basic steps involve simplifying the expression so that we can isolate the variable on one side. Always maintain the inequality's direction unless you multiply or divide by a negative number, which reverses the inequality symbol. Keep in mind:

• When solving, ensure the variable ends up on one side.
• Perform the same operation on both sides to keep the inequality balanced.
• Check your solution by substituting values back into the original context.

Working with Negative Values

Negative values occur when a result is less than zero. When solving inequalities or equations, understanding how negative values interact with operations is crucial. Take the inequality \(x - 7 < 0\). Our task is to determine when this expression yields a value less than zero. Here's how you tackle negative values:

• Subtracting a number can lead to negative results; here, 7 is subtracted from \(x\).
• If \(x\) is less than 7, \(x-7\) becomes negative.
• Visualize on a number line: values to the left of 7 make \(x-7\) negative.

Negative values are key in many algebraic scenarios, where they change how inequalities behave. Remember, while adding or subtracting numbers doesn't flip the inequality sign, multiplying or dividing by a negative does.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Expressions can be simple like \(x - 7\) or complex with multiple terms. The beauty of algebra lies in understanding the relationships they represent.When dealing with an expression such as \(x - 7\), you're essentially considering an unknown value, \(x\), and adjusting it by 7. Expressions set the stage for creating equations and inequalities, which can then be solved.Here's what you should remember about these expressions:

• Terms like \(x\) represent variables that can take various values.
• Constants like \(7\) are fixed numbers within the expression.
• They often form the basis of functions, equations, or inequalities.

Simplifying expressions often involves combining like terms and applying distributive properties. Your aim is to transform expressions for easy understanding and use in problem-solving contexts.