Problem 3 - Algebra 1: Concepts and Skills

Question

Determine whether the inequality is a multi-step inequality. Then explain how you would solve the inequality. $$ d+2>-1 $$

Step-by-Step Solution

Answer

The inequality \(d+2 > -1\) is a single-step inequality. The solution is \(d > -3\).

Step 1: Identify type of inequality

Upon inspection, it can be concluded that \(d + 2 > -1\) is a single-step inequality because it only involves one operation - namely, the addition of 2 to 'd'.

Step 2: Solve the inequality

The goal is to isolate 'd' and to do this, subtract 2 from both sides of the inequality. As a result, we get \(d > -1 - 2\).

Step 3: Simplify the inequality

Perform the subtraction on the right side of the inequality, which gives \(d > -3\).

Key Concepts

Single-Step Inequalities

When dealing with inequalities, understanding the concept of single-step inequalities is crucial. A single-step inequality, such as the example problem provided, involves just one operation to solve. In the given inequality, \(d + 2 > -1\), there is only the operation of adding 2. To solve it, the opposite of the addition operation needs to be performed – which is subtraction.

To isolate the variable and solve this single-step inequality, you would simply subtract 2 from both sides of the inequality. This operation will balance out the addition of 2 on the left side, leaving the variable 'd' alone. The result after performing this subtraction is the simplified form of the inequality, which shows the range of values 'd' can take to satisfy the original inequality (\(d > -3\)). Always check the original inequality to confirm that the values make sense.

Multi-Step Inequalities

In contrast to single-step inequalities, multi-step inequalities involve more than one operation to isolate the variable. Such inequalities may include a combination of addition, subtraction, multiplication, division, or even variables on both sides of the inequality sign. To solve these, we follow a sequence of steps to simplify the inequality until we ultimately isolate the variable.

For instance, if we had an inequality like \(3x - 5 > 7\), we would first add 5 to both sides to eliminate the subtraction, rendering \(3x > 12\), and then we'd divide both sides by 3 to isolate x, resulting in \(x > 4\). During the process, it's essential to remember that if we multiply or divide both sides of the inequality by a negative number, we must reverse the inequality sign to maintain the inequality's truth.

Isolate the Variable

Isolating the variable is the foundational step in solving both single-step and multi-step inequalities. This means manipulating the inequality to get the variable by itself on one side of the inequality sign. The variable gets isolated by performing operations that undo whatever is being done to the variable. These operations must be done on both sides to keep the inequality balanced.

When isolating variables, it's also crucial to pay attention to the inequality's direction. It will stay the same when adding or subtracting on both sides. However, if you multiply or divide by a negative number, this will flip the inequality's direction. For example, if we multiply both sides of an inequality by -1, a 'greater than' sign (>) will become a 'less than' sign (<), and vice versa.