Question

Which of the following operations could be performed on both sides of the inequality \(-2 x>4\) to require the direction of the inequality sign be changed while keeping \(x\) on the left-hand side of the inequality? (A) Add 4 (B) Subtract 7 (C) Divide by -2 (D) Multiply by 12

Short answer

(C) Divide by -2.

Step-by-step solution

Identify the Original Inequality

The given inequality is \(-2x > 4\). Our goal is to manipulate the inequality while keeping \(x\) on the left-hand side and potentially changing the direction of the inequality sign only under certain conditions.

Rule for Changing Inequality Direction

The direction of an inequality reverses when both sides are multiplied or divided by a negative number. We will apply this rule to the given inequality to see which operation will change its direction.

Evaluate Option A: Add 4

Adding 4 to both sides of the inequality \(-2x > 4\) results in \(-2x + 4 > 8\). The inequality's direction remains unchanged because merely adding a number does not change the inequality's direction.

Evaluate Option B: Subtract 7

Subtracting 7 from both sides results in the inequality \(-2x - 7 > -3\). Again, the direction of the inequality remains unchanged since subtraction of a number does not affect the inequality's direction.

Evaluate Option C: Divide by -2

Dividing both sides of \(-2x > 4\) by \(-2\) results in the inequality \(x < -2\). Notice here that the direction of the inequality sign has changed because we divided by a negative number.

Evaluate Option D: Multiply by 12

Multiplying both sides of the inequality \(-2x > 4\) by 12 gives \(-24x > 48\). The direction of the inequality does not change because multiplying by a positive number does not result in a reversal of the inequality's direction.

Key concepts

Inequality Direction

When working with inequalities, it's essential to understand how the direction of the inequality sign may change based on the mathematical operations applied to both sides. An inequality's direction, indicated by signs like \( > \), \( < \), \( \geq \), or \( \leq \), shows the relationship between two expressions. When you perform certain operations, the direction of these signs may remain the same or change.

• Adding or subtracting a number from both sides of the inequality does not change its direction. This means if you have an inequality like \( x > y \), adding or subtracting any number keeps it as \( x > y \).
• Multiplying or dividing both sides of an inequality by a positive number also keeps the inequality's direction unchanged.
• However, multiplying or dividing both sides by a negative number reverses the direction of the inequality. This critical rule ensures the mathematical relationship held by the inequality is maintained.

Remembering these rules helps avoid mistakes when solving inequalities or when deciding how to manipulate them while ensuring that the solution set stays valid.

Negative Numbers

Negative numbers have unique properties that can affect operations in inequalities. Understanding these effects is crucial for correctly managing inequality direction changes.

• Negative numbers are less than zero, meaning they lay on the opposite side of positive numbers on the number line. For example, –3 is to the left of 1 on a number line.
• The crucial point where negative numbers play a significant role in inequalities is when multiplying or dividing an inequality by a negative number. For example, if you have the inequality \(-2x > 4\), dividing by a negative flips the inequality sign, resulting in \(x < -2\).
• Use caution: inadvertently ignoring this rule can lead to incorrect solutions. Always double-check when you've used a negative number to manipulate an inequality.

Working with negative numbers is foundational in algebra, so understanding their properties ensures you solve inequalities precisely and confidently.

Algebraic Manipulation

Algebraic manipulation is a key skill in solving inequalities. It involves performing operations to both sides of an inequality to isolate the variable, often while maintaining the equality or reversing its direction if necessary.

• Begin by identifying the operation needed to simplify the inequality. This typically involves isolating the variable to find its possible values.
• Use inverse operations to simplify. For instance, if a variable is multiplied by a factor, you can divide by that factor to simplify. The same applies to addition or subtraction—use the opposite operation to simplify.
• Caution is crucial: when a variable is associated with a negative coefficient, remember the rule about inequality direction—multiplying or dividing by this coefficient will reverse the inequality sign.

For example, with the inequality \(-2x > 4\), algebraic manipulation by dividing both sides by \(-2\) results in the inequality \(x < -2\). These manipulations reveal that \(x\) must be less than \(-2\), showcasing the necessity of understanding algebraic processes to find the correct solution. Thus, mastering algebraic manipulation is crucial for solving inequalities effectively.