Problem 30 - Algebra 2

Question

Define a variable and write an inequality for each problem. Then solve. The product of \(-4\) and a number is at least \(35 .\)

Step-by-Step Solution

Answer

The solution is \( x \leq -8.75 \).

Step 1: Define the Variable

Let's start by defining the variable. We'll let \( x \) represent the number we are trying to find.

Step 2: Write the Inequality

The problem states that the product of \(-4\) and a number is at least \(35\). This is represented by the inequality: \(-4x \geq 35\).

Step 3: Solve for the Variable

To isolate \( x \), divide both sides of the inequality by \(-4\). Remember, dividing by a negative number reverses the inequality sign: \( x \leq -\frac{35}{4} \).

Step 4: Simplify the Solution

Simplify \(-\frac{35}{4}\) to a decimal: \( -\frac{35}{4} = -8.75 \). Thus, the solution to the inequality is \( x \leq -8.75 \).

Key Concepts

Defining Variables

When tackling mathematical problems, especially inequalities, it's crucial first to define variables clearly. A variable is a symbol, usually a letter, used to represent an unknown value. In our exercise, we define the variable as follows:

• Let \( x \) represent the unknown number.

Choosing the right variable is mostly about simplicity and clarity, making the equation easier to handle and understand. This step sets the foundation for writing and solving the inequality accurately. Without properly defining the variable, the subsequent steps might not make logical sense.

Solving Inequalities

Solving inequalities involves finding the set of values for the variable that make the inequality true. This process is similar to solving equations, with a few added rules—primarily, the rule concerning inequality sign reversal.

Steps to Solve an Inequality

• Write the Inequality: Start by expressing the condition using a mathematical inequality. In our case, "The product of \(-4\) and a number is at least \(35\)" becomes \(-4x \geq 35\).
• Isolate the Variable: To solve for \( x \), divide both sides by \(-4\). Remember to reverse the inequality when dividing by a negative: \( x \leq -\frac{35}{4} \).
• Simplify: Although \(-\frac{35}{4}\) is mathematically correct, it's often helpful to simplify it to a decimal, \(-8.75\), for easier interpretation.

Understanding these steps ensures that you effectively solve inequalities and interpret the solutions correctly within the context of a problem.

Multiplication of Integers

Multiplication of integers is a fundamental arithmetic operation that applies in various mathematical contexts, including inequalities. Understanding how multiplication interacts with negative numbers is crucial.

Key Concepts of Integer Multiplication

• Sign Rules: The product of two integers with the same sign (both positive or both negative) is positive. The product of integers with different signs is negative.
• Example in Context: Here, multiplying \(-4\) by a positive or negative integer affects the inequality directly. This operation is represented within the inequality \(-4x \geq 35\). It's essential to apply the sign rule correctly when manipulating inequalities.

Recognizing the impact of multiplication is pivotal when solving inequalities, as it determines not just the magnitude but also the direction of the inequality.