Problem 87 - Algebra 1: Concepts and Skills

Question

Solve the inequality. \(3-x>-4\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 7\).

Step 1: Rearrange the inequality

We can start by subtracting 3 from both sides of the inequality to isolate \(x\): \n \(3 - x - 3 > -4 - 3\). This simplifies to \(-x > -7\).

Step 2: Multiply by -1

Since multiplying by -1 leads to a flip in the direction of the inequality sign, we can multiply both sides of the inequality by -1 to further isolate \(x\): \(-1(-x) < -1(-7)\). This simplifies to \(x < 7\).

Key Concepts

Algebraic Inequalities

Inequalities are a fundamental concept in algebra that express the relationship between two values where one is not necessarily equal to the other. They indicate that one value is greater than, less than, greater than or equal to, or less than or equal to another value. Unlike equations, inequalities do not show exactness but rather a range of possible solutions. For example, the inequality \(3-x>-4\) suggests that there are many numbers that \(x\) can be, as long as they satisfy the condition of being less than 7 after manipulation.

The process of solving an inequality is similar to that of solving an equation, with one critical difference; when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign flips. This property is crucial to remember, as it is a common source of mistakes when students are learning to solve inequalities.

Inequality Manipulation

Manipulating inequalities effectively is key to finding their solutions. This process can involve several steps, which usually include adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. One must be particularly careful with multiplication or division by negative numbers, as this operation reverses the direction of the inequality sign.

For example, if we have \(-x > -7\), and we wish to isolate \(x\), we multiply both sides by -1 to get \(x < 7\). It's essential to note that the inequality sign reversed from '>' to '<' as a result of multiplying by a negative. Understanding and remembering this rule is vital for correctly manipulating inequalities and avoiding common errors.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are the building blocks for solving algebraic problems, including inequalities. When we solve an inequality, we perform these operations with the goal of isolating the variable on one side of the inequality sign. It's important these operations are carried out equally on both sides to maintain the balance of the inequality.

In the given problem, we start by subtracting 3 from both sides, an example of a straightforward operation that doesn't affect the inequality's direction. It's only when we perform the operation of multiplying by -1 that we need to be mindful of the rule regarding the direction change of the inequality. Regular practice with these operations will strengthen one's ability to solve inequalities with confidence and accuracy.