Question

Fill in the blank to form a correct inequality statement. If \(x \leq-4,\) then \(-3 x________12\)

Short answer

\( -3x \geq 12 \).

Step-by-step solution

- Understand the Given Inequality

The initial inequality given is: \[ x \leq -4 \].

- Substitute the Value

Since \( x \leq -4 \), substitute \(x = -4\) into the expression \( -3x \) to check for the boundary condition. We get: \[ -3(-4) = 12 \].

- Formulate the Correct Inequality

From Step 2, we found that when \( x = -4 \), the expression \( -3x \) equals 12. Since the inequality \( x \leq -4 \) allows \( x \) to take any value less than or equal to -4, \(-3x \) will be greater than or equal to 12. Therefore, \[ -3x \geq 12 \].

Key concepts

inequality operations

To understand inequality operations, it's important to remember that inequalities are like equations, but with relationships such as >, <, ≥, and ≤. Here, we're given the inequality:
$$ x eq -4 $$
This tells us that x can be -4 or any value smaller. To find out how this affects other parts of an equation, we'll manipulate or transform it. These transformations must preserve the inequality's truth. For instance, multiplying both sides of an inequality by a negative number reverses the inequality sign. In this case:
$$ -3x $$. We use multiplication to see its effect on the inequality, maintaining proper inequality operations.

boundary condition

The boundary condition tells us the critical points in an inequality. It is the value where the inequality switches from true to false or vice versa. In our example, the boundary condition is
$$ x = -4 $$.
We substitute this value to find out the result of the expression
$$ -3x \text{. When } x = -4 $$,
we get:
$$ -3(-4) = 12. $$
Any deviation towards values smaller than -4 changes the expression's value. This makes it easier to evaluate which side of the inequality each value will fall on, helping us form a correct inequality.

substitution

Substitution in mathematics means placing a specific value or expression into an equation or inequality. Here, we substitute
$$ x \text{ with } -4$$.
When we do:
$$ -3x$$ becomes \(12\).
We used substitution to verify how the expression
$$ -3x $$
behaved at a critical or boundary value. This step helps establish the necessary relationship, leading to the determination that the complete inequality is:
$$ -3x \text{ greater than or equal to 12 } $$
This process of substitution is vital in mathematics to verify solutions, especially in inequalities where multiple values can change the result.