Problem 1 - College Algebra

Question

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

Step-by-Step Solution

Answer

The inequality sign flipped because we divided by a negative number.

Step 1: Start with Original Inequality

We begin with the inequality given in Step 1: \(-2x > 6\). This inequality states that twice the negative of the variable \(x\) is greater than 6.

Step 2: Isolate Variable by Division

To isolate \(x\), we need to divide both sides of the inequality by -2. Remember, when dividing or multiplying an inequality by a negative number, the direction of the inequality sign changes. Thus, dividing both sides by -2 gives us:\[x < \frac{6}{-2} = -3\]This changes the inequality direction from \(>\) to \(<\).

Key Concepts

Inequality Division

Inequality division is a crucial step in solving inequalities where our main goal is to isolate the variable. In our original exercise, we started with the inequality \(-2x > 6\). Here, the variable \(x\) is multiplied by \(-2\). To solve for \(x\), we needed to "undo" this operation by using division. Dividing both sides of the inequality by \(-2\) allowed us to isolate \(x\) on one side of the inequality. Keep in mind:

• Always perform the same operation on both sides of the inequality.
• Remember to consider special rules, such as changing the inequality sign when dividing by a negative number.

Dividing each side by \(-2\) gave us \(x < -3\). Notice how we not only isolated \(x\) but also demonstrated proper inequality division.

Solving Inequalities

Solving inequalities involves finding a range or a specific set of values for the variable that makes the inequality true. The process shares similarities with solving equations, such as:

• Combining like terms if necessary.
• Applying inverse operations to both sides to isolate the variable.

However, solving inequalities differs slightly from solving equations due to the potential to reverse the inequality sign when dealing with negative coefficients, as demonstrated in our exercise. A solution in inequalities doesn't just satisfy at a single point; instead, it identifies a range of values. In our case, the inequality \(x < -3\) means all values less than \(-3\) satisfy the inequality.

Inequality Sign Change

The inequality sign change is a vital rule to remember when solving inequalities. This change occurs whenever you multiply or divide both sides of an inequality by a negative number. This is because multiplying or dividing by a negative flips the direction of inequality because it effectively reverses the direction of the number line. In the exercise, upon dividing both sides by \(-2\), the 'greater than' sign \(>\) shifted to a 'less than' \(<\) sign, resulting in the solution \(x < -3\).

• Always remember to change direction when multiplying or dividing by negatives.
• Review your final result to confirm the inequality's direction aligns with these operations.

This rule can be tricky at first, but with practice, becomes a natural part of solving inequalities.