Problem 1 - College Algebra with Corequisite Support

Question

When solving an inequality, explain what happened from Step 1 to Step 2: Step \(1 -2 x >6\) Step 2 \( x < -3\)

Step-by-Step Solution

Answer

Divided by \(-2\) and flipped the inequality sign.

Step 1: Identify the Inequality Equation

The inequality given is \(-2x > 6\). The goal is to solve for \(x\), meaning we want \(x\) on one side of the inequality by itself.

Step 2: Isolate the Variable

To isolate \(x\), we need to get rid of the coefficient \(-2\) multiplied by \(x\). To do this, we divide each side of the inequality by \(-2\). It is important to remember that when we divide or multiply an inequality by a negative number, the inequality sign flips direction. Thus, \(-2x > 6\) becomes \(x < -3\).

Key Concepts

Inequality Reversal

One of the most crucial things to remember when solving inequalities is that the inequality sign must be reversed when both sides are multiplied or divided by a negative number. This is different from solving equations and is a distinctive characteristic of inequalities.
An easy way to remember is to think about the number line:

• When you multiply or divide by a positive number, the order or direction of numbers doesn't change.
• However, multiplying or dividing by a negative flips the order.

This reversal occurs because what was once smaller can become larger and vice versa. For example,

Example

If \(-1 < 2\), multiplying by \(-1\) yields \(1 > -2\), showing how the inequality flips. Thus, when solving \(-2x > 6\), dividing by \(-2\) reverses the inequality to \(x < -3\). Always remember to flip the inequality sign in these situations. It is a critical step in proper inequality manipulation.

Isolating Variables

Isolating variables is a core step in solving algebraic equations and inequalities. When we isolate a variable, we aim to have it by itself on one side of the inequality with no coefficients or constants attached. This process helps us determine the solution.
In the exercise you have already identified your inequality as \(-2x > 6\). Here, the goal is to solve for \(x\). To achieve this, we need to eliminate any numbers present with \(x\) on its side of the inequality.

How To Isolate

• Identify the coefficient of the variable, which in this case is \(-2\).
• Eliminate the coefficient by performing the inverse operation, so divide both sides of the inequality by \(-2\).

Thus, \(-2x > 6\) becomes \(x < -3\). By carefully isolating the variable, we simplify the inequality, making it straightforward to identify the solution.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging equations or inequalities to simplify or solve them. This involves operations such as addition, subtraction, multiplication, or division to both sides of an inequality or equation. This is key in mathematics, providing methods to solve for unknowns.
With inequalities, there are specific rules to maintain their properties, such as inequality reversal, as previously described.

Key Steps in Manipulation

• Identify terms involving your variable and constants.
• Perform inverse operations to begin simplifying each side.
• Be vigilant about inequality direction changes, especially multiplying/dividing by negatives.

For the problem \(-2x > 6\), divide by \(-2\) to achieve \(x < -3\). This manipulation not only isolates the variable but also ensures the inequality correctly reflects the relationship between \(x\) and its possible values. Precise algebraic manipulation leads to clear and correct solutions.