Problem 86 - Algebra 1: Concepts and Skills

Question

Solve the inequality. \(-x-2<-5\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 3\).

Step 1: Add 2 to both sides of the inequality

We start by adding 2 to both sides of the inequality to cancel out the -2 on the left. This leaves us with \(-x <-5 + 2\) which simplifies to \(-x < -3\).

Step 2: Multiply both sides of the inequality by -1

Since our variable x is preceded by a negative sign, we need to get rid of this to get x by itself. We do this by multiplying both sides of the inequality by -1, which causes the inequality to flip in the opposite direction. So we get \(x > 3\). This is the final answer.

Key Concepts

Inequality Notation

Understanding inequality notation is crucial when solving mathematical inequalities. Inequality symbols, such as '<', '>', 'leq', and 'geq', are used to express the relationship between two expressions. For instance, 'a < b' means that 'a' is less than 'b'.

When dealing with inequalities, it's important to remember that these symbols define not just a single number but a range of possible values. For example, when we solve for a variable and find 'x > 3', this means that 'x' can be any number greater than 3, including numbers like 4, 5, and even 3.01.

Two-Step Inequalities

Two-step inequalities are algebraic problems that require two steps to isolate the variable and solve the inequality. Similar to two-step equations, they often involve an additional term and a coefficient connected to the variable.

To solve these inequalities, one must perform inverse operations to eliminate these two barriers. First, you would typically add or subtract to remove the constant term, and then you would multiply or divide to isolate the variable. It's crucial throughout this process to maintain the balance of the inequality, performing identical operations on both sides.

Multiplying by Negative Numbers

When multiplying both sides of an inequality by a negative number, a key rule must be followed: the direction of the inequality must be reversed. This is a critical concept in algebra that prevents errors in finding the solution set.

The reason behind this rule lies in the number line's nature. For any two numbers 'a' and 'b', if 'a' is less than 'b', multiplying both by a negative number reverses their positions on a number line – making 'a' greater than 'b'. Therefore, when the inequality (-x < -3) is encountered, and we multiply by -1 to isolate 'x', we get 'x > 3', ensuring the solution reflects the correct set of numbers that satisfy the original inequality.