Problem 39 - Prealgebra

Question

The product of an integer and \(-7\) is less than \(-84 .\) Find the least integer that meets this condition.

Step-by-Step Solution

Answer

The least integer that meets the condition is 13.

Step 1: Create the inequality

First, we need to set up an inequality based on the given information. We are told that the product of an integer \( x \) and \( -7 \) is less than \( -84 \). This can be written as: \(-7x < -84\).

Step 2: Solve the inequality

Next, we solve the inequality for \( x \). To isolate \( x \), divide both sides of the inequality by \( -7 \). Remember, dividing or multiplying an inequality by a negative number reverses the inequality sign. This gives us:\[ x > \frac{-84}{-7} \]Simplifying the division on the right-hand side, we get:\[ x > 12 \]

Step 3: Identify the least integer solution

The inequality \( x > 12 \) means that \( x \) can be any number greater than 12. The smallest integer greater than 12 is 13.

Key Concepts

Understanding Integer Solutions

Integers are whole numbers that can be positive, negative, or zero. When solving problems involving integers, identifying the possible whole numbers that satisfy certain conditions is key. In the context of inequalities, the integer solutions are those whole numbers that make the inequality statement true. For instance, if given an inequality

• Manually test integers in the solution set to confirm they meet the condition.
• Scan the range of possible integer values without the confusion of fractions or decimals.

In the exercise, solving for the smallest integer greater than 12 makes it easy to find the first integer solution: 13. This direct approach is central when working with linear inequalities that call for integer solutions.

The Process of Solving Inequalities

Solving linear inequalities is akin to solving linear equations, but there's a crucial difference when it involves multiplying or dividing by negative numbers. The inequality sign (

• Write down the inequality correctly from the problem description.
• Perform operations to isolate the variable, such as addition, subtraction, multiplication, or division.
• Switch the inequality sign when multiplying or dividing by a negative number.

In our example, solving

• Divide both sides by -7.
• Reverse the inequality sign due to the division by a negative.

These steps ensure a correct and valid solution of inequalities in any algebra problem.

Working with Negative Numbers

Negative numbers often appear in equations and inequalities, representing values below zero. Their rules differ from positive numbers particularly in multiplication and division.

• When multiplying or dividing two negative numbers, the result is positive.
• Reversing signs in inequalities is crucial when dividing or multiplying across negative numbers.

In our exercise, the inequality involves

• Recognize that
• Convert the result of solving the inequality into an accessible integer solution.

Handling negative numbers and understanding their properties are essential skills for solving algebraic problems effectively.