Problem 102 - Algebra 1

Question

Solve the inequality. $$-x-2<-5$$

Step-by-Step Solution

Answer

The inequality \( -x-2<-5 \) is solved as \( x>3 \)

Step 1: Simplify the Inequality by Adding or Subtracting

To isolate 'x', begin by adding 2 to both sides of the inequality. This simplifies to: \[ -x < -5 + 2 \],which is,\[ -x < -3 \]

Step 2: Dividing by a Negative Number

Next, divide both sides of the inequality by -1. Remember that when you divide both sides of an inequality by a negative number, the direction of the inequality sign changes.This gives: \[ x > 3 \]

Key Concepts

Inequality Manipulation

Understanding how to manipulate inequalities is fundamental in algebra. Similar to solving an equation, solving an inequality involves performing operations to isolate the variable. However, there's a crucial difference: the inequality sign may change direction depending on the operation.

For instance, when you add or subtract the same number from both sides, the inequality's direction remains unchanged. This means if you have \(-x - 2 < -5\), adding 2 to both sides yields \(-x < -3\), with the inequality sign pointing the same way. The goal is always to isolate the variable, and throughout this process, maintaining a balance on both sides is key.

Dividing by a Negative Number

Dividing by a negative number is a critical operation in solving inequalities and it has a special rule. The rule states that whenever you divide (or multiply) both sides of an inequality by a negative number, the inequality sign must be flipped.

In the given problem, when you get to \(-x < -3\), to solve for 'x', you divide by -1 leading to \(x > 3\), noticing that the sign changed from '<' to '>'. This is because dividing by a negative number technically reverses the order of the values on the number line, thus changing the inequality's sense of direction.

Linear Inequalities

Linear inequalities, much like linear equations, represent the relationship between two expressions that are not necessarily equal but have a 'greater than', 'less than', 'greater than or equal to', or 'less than or equal to' relationship. They form a linear graph when plotted on a coordinate plane but, unlike a linear equation's single line, represent a region.

When you solve a linear inequality such as \(-x - 2 < -5\), you're finding all the values of 'x' that make the inequality true. These values form a continuous range and can be expressed in several ways: using inequality notation, a number line, or interval notation, and each method shows the solution set's range clearly.

Algebraic Steps

Approaching algebraic problems involves a series of steps, each serving a purpose to simplify and solve the given problem methodically. The progression from the original inequality to the solution involves simplification and the application of algebraic rules. For the problem \(-x - 2 < -5\), the initial step is to get rid of the constant term by adding 2 to both sides, resulting in \(-x < -3\).

Next, to isolate 'x', you divide by -1, and crucially, flip the inequality sign to maintain the truth of the inequality. With \(x > 3\), 'x' is now isolated, and the inequality is solved, reflecting the algebraic steps taken to manipulate the expression into its simplest form.