Question

Fill in the blank to form a correct inequality statement. If \(x>-2,\) then \(-4 x____8\)

Short answer

-4x < 8

Step-by-step solution

- Analyze the given inequality

Start by analyzing the given inequality. We have the inequality \(x > -2\).

- Apply the inequality to the expression

Since we need to form an inequality for \(-4x\), first, we need to multiply both sides of the inequality \(x > -2\) by -4.

- Reverse the inequality sign

Remember, when we multiply or divide both sides of an inequality by a negative number, the inequality sign reverses. Therefore, multiplying both sides by -4 gives us: \(-4x < -4(-2)\)

- Simplify the expression

Simplify the expression on the right side: \(-4(-2) = 8\) So, the inequality becomes: \(-4x < 8\)

Key concepts

inequality manipulation

Inequality manipulation is a crucial aspect of solving problems in algebra. When we talk about inequality manipulation, we are typically referring to performing operations like addition, subtraction, multiplication, or division on the inequality to isolate the variable. The goal is often to find a solution range for the variable that satisfies the given inequality.

For instance, if we have the inequality \(x > -2\), we can perform several valid operations:

• Add or subtract a constant from both sides.
• Multiply or divide both sides by a positive number without changing the direction of the inequality sign.

In our exercise, we encountered the inequality \(x > -2\) and manipulated it by multiplying both sides by -4 to begin isolating \(-4x\).
Understanding how to manipulate inequalities properly is essential for solving more complex algebraic expressions.

multiplying inequalities

Multiplying inequalities is a bit tricky because special rules apply when dealing with negative numbers. When you multiply both sides of an inequality by a positive number, the direction of the inequality sign does not change. For example, if \(a < b\), and we multiply both sides by 2, it remains \(2a < 2b\).

However, when you multiply both sides by a negative number, the inequality sign must be reversed. Consider the inequality \(x > -2\). If we multiply both sides by -4, we reverse the sign to maintain a true statement:

\[ -4x < -4(-2) \]
This is due to the properties of numbers and how they behave under multiplication. Always be careful and remember this rule when working with inequalities involving negative numbers!

reversing inequality signs

Reversing the inequality sign is necessary when you multiply or divide both sides of an inequality by a negative number. This is a fundamental rule in algebra and helps ensure the inequality remains valid. Let's take a closer look at why this is important.

For example, consider the inequality \(x > -2\). When we multiply by -4, we have:

\[ x > -2 \implies -4x < -4(-2) \]
Simplifying the right side, we get: \[-4x < 8 \]
Without reversing the inequality sign, the statement would be incorrect. Understanding this rule prevents common mistakes and ensures you arrive at the correct solutions. As a tip:

• Always double-check your work for sign reversals when multiplying or dividing by negative numbers.