Question

Fill in the blank to form a correct inequality statement. If \(x \leq 3,\) then \(2 x______6\)

Short answer

2x \leq 6

Step-by-step solution

Substitute the Value

Substitute the maximum possible value of the variable into the expression. Since the condition is given as \( x \leq 3 \), use \( x = 3 \).

Multiply the Values

Now, substitute \( x = 3 \) into the expression \( 2x \). This gives \( 2 \times 3 = 6 \).

Form the Inequality

Since \( x \) can take any value less than or equal to 3, \( 2x \) will be less than or equal to 6. Therefore, the correct inequality is \( 2x \leq 6 \).

Key concepts

Substitute the Value

The first step in forming an inequality is to substitute the variable with its maximum possible value. Here, the condition is given as \(x \leq 3\). This signifies that \(x\) can be any number from negative infinity up to and including 3. To proceed, we substitute \(x\) with its maximum value of 3 for simplicity. This substitution helps us simplify and understand the inequality better. Always remember to substitute values carefully to avoid errors. Using the maximum value ensures that we are working with the largest potential outcome within the given range.

Multiply the Values

After successfully substituting the value, we now move to multiply it within the expression. In this problem, the equation given is \(2x\). When we substitute \(x = 3\), it transforms into \(2 \cdot 3\). Performing the multiplication gives us \(2 \times 3 = 6\). Multiplication steps are straightforward but crucial, as any small mistake can alter the entire inequality. Remember, always perform operations step by step, especially when dealing with multiple terms or higher algebraic expressions, ensuring precision and accuracy in every step.

Form the Inequality

The final step is to form the inequality statement. After substituting and multiplying, we have \(2x = 6\) when \(x = 3\). Knowing that \(x\) can also be less than 3, implies the product \(2x\) will be less than or equal to 6. Therefore, we conclude that the inequality for \ x\ \(is\) 2\text(\text )\(2 \). Remember, inequalities represent a range of values, not just a single outcome. That is why the inequality encompasses all potential values less than or equal to those derived through substitution and multiplication steps.