Question

Fill in the blank to form a correct inequality statement. If \(x \geq-4,\) then \(3 x_____-12\)

Short answer

3x \geq -12

Step-by-step solution

Understand the given inequality

We are given the inequality \(x \geq -4\) and need to determine the relationship between \(3x\) and \(-12\).

Multiply the inequality by 3

To maintain the inequality, multiply every term by 3: \(3x \geq 3 \times -4\).

Simplify the inequality

Simplify the right-hand side: \(3x \geq -12\).

Fill in the blank

By comparing the given statement \(3x \_ \_ \-12\) and the simplified inequality \(3x \geq -12\), we can fill in the blank with the appropriate inequality symbol.

Key concepts

Multiplying inequalities

When you multiply an inequality by a positive number, the direction of the inequality sign remains unchanged. This is because the relationship between the numbers does not change. However, if you multiply by a negative number, you need to reverse the inequality sign. This is an important rule in algebra and helps maintain the true value of the expression.

In the provided exercise, we start with the inequality \( x \geq -4 \). We need to determine the relationship between \( 3x \) and \( -12 \). To do this, we multiply both sides of the inequality by 3:

- Given: \( x \geq -4 \)
- Multiply both sides by 3: \( 3x \geq 3 \times -4 \)
- Simplify: \( 3x \geq -12 \)

In this example, we multiplied by a positive number, so the direction of the inequality did not change. This is crucial for maintaining the correct relationship between \( 3x \) and \( -12 \).

Simplifying inequalities

Simplifying inequalities involves performing basic arithmetic operations to both sides of the inequality to make it easier to understand or solve. The goal is to isolate the variable on one side of the inequality sign. This process often includes addition, subtraction, multiplication, or division.

In the given exercise, after multiplying the original inequality by 3, we have \( 3x \geq -12 \). This inequality is already simplified, as the variable \( x \) is isolated on the left-hand side.

The steps for simplifying might generally involve:

• Combining like terms.
• Using the distributive property.
• Performing inverse operations (e.g., if there is addition, subtract the same value from both sides).

For our example, after multiplying, we do not need additional steps, as the inequality \( 3x \geq -12 \) is already in its simplest form.

Algebraic expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the context of inequalities, they involve expressions where one side is compared to the other using inequality symbols like \( \geq \), \( \leq \), \( > \), or \( < \).

In the given exercise, we start with the expression \( x \geq -4 \). After multiplying both sides by 3, the inequality becomes \( 3x \geq -12 \).

Key points to remember about algebraic expressions include:

• They can represent a wide range of situations and quantities.
• Proper manipulation, including using operations like multiplication, helps in solving or simplifying them.
• Inequality symbols are crucial in defining the relationship between expressions.

In our exercise, knowing how to handle the algebraic expressions and apply the rules of inequalities helps us fill in the blank correctly to form \( 3x \geq -12 \). This demonstrates the relationship between the variable \( x \) and the number -12 when multiplied by 3.