Problem 92 - Introductory Algebra for College Students

Question

Use properties of inequality to rewrite each inequality so that \(x\) is isolated on one side. \(-2 x-a \leq b\)

Step-by-Step Solution

Answer

The solution to the inequality \(-2 x-a \leq b\) where \(x\) is isolated on one side is \(x \geq -\frac{b + a}{2}\).

Step 1: Move the constant term to the right side of the inequality.

In order to isolate \(x\), let's first remove the constant \(a\) on the left side of the inequality. This is done by adding \(a\) to both sides of the inequality, leaving us with \(-2x \leq b + a\).

Step 2: Isolate \(x\) on the left side of the inequality.

The term involving \(x\) is \(-2x\). To isolate \(x\), we can divide through by \(-2\), but remember that dividing by a negative number flips the inequality sign, thus giving the final inequality \(x \geq -\frac{b + a}{2}\). Remember that the negative sign is moved with \(b + a\) to make the denominator positive.

Step 3: Verify the inequality.

The final step is to check if the solution falls within the parameters given by the original inequality. Assuming that \(b\) and \(a\) are any real numbers and recall that the direction of the inequality sign was flipped when dividing by -2, we can see that the solution provided will indeed satisfy the initial inequality \(-2 x-a \leq b\). Hence, the solution is correct.

Key Concepts

Understanding Properties of Inequality

Learning to work with inequalities is a foundational skill in algebra. Inequalities describe the relative size of two values and come with rules, much like equations, but with some key differences. These rules are known as the properties of inequality. When solving an inequality like \( -2x - a \leq b \), remembering these properties is essential to ensure accurate solutions.

First, the addition property of inequality allows us to add or subtract the same number from both sides of the inequality without altering the inequality's direction. For example, by adding \(a\) to both sides of \( -2x - a \leq b \), we preserve the inequality while re-arranging it into a simpler form. This step is often required to isolate the variable. Second, the multiplication and division properties of inequality state that multiplying or dividing both sides of an inequality by a positive number will not affect the direction of the inequality. That is, if \( a > b \), then \( ac > bc \) provided that \( c > 0 \). However, inversely multiplying or dividing by a negative number flips the inequality's direction due to the inverse properties of numbers, which is crucial when isolating variables that are multiplied by negative values.

Isolating Variables in Inequalities

When tackling an inequality, our primary goal is often to isolate the variable—much like in a standard equation— so that we have the variable on one side and everything else on the other. To achieve this in the given inequality \( -2x - a \leq b \), the process involves two main steps.

Initially, we address any constants that are on the same side as the variable. As with our example, we need to remove \( -a \) by performing the inverse operation, which is adding \(a\) to both sides. This results in \( -2x \leq b + a \), bringing us one step closer to isolating \( x\).

Moving to the Coefficient

Afterwards, we focus on the coefficient next to \( x \). Since \( x \) is multiplied by -2, we need to perform the opposite operation, which in this case is division. But be watchful! Since the coefficient is negative, this action will influence the direction of the inequality, leading us into our next concept, the 'inequality sign flipping'. Ensuring variables are isolated correctly is vital for accurately representing the solution set of an inequality.

The Rule of Inequality Sign Flipping

Inequality sign flipping is a critical concept when solving inequalities, especially when a negative number comes into play. It's a rule that can cause confusion but mastering it is crucial for correct solutions.

Put simply, if we multiply or divide both sides of an inequality by a negative number, the direction of the inequality changes. This is the opposite of what happens with multiplication and division by positive numbers. Let's revisit our example from the second step of solving \( -2x - a \leq b \). When we divide both sides by -2 to isolate \( x\), the \leq symbol flips to become \geq, resulting in \( x \geq -\frac{b + a}{2} \). This flipping ensures that the relationship between the sides remains true in the context of a negative divisor or multiplier.

Verifying for Accuracy

It's good practice to test your solution by plugging in numbers to verify the inequality holds. If the sign is flipped appropriately, the inequality will consistently work, reaffirming that the rules of inequality have been correctly applied.