Question

Fill in the blank to form a correct inequality statement. If \(-\frac{1}{4} x>1,\) then \(x______-4\)

Short answer

x < -4

Step-by-step solution

- Identify the given inequality

The given inequality is \[\begin{equation}-\frac{1}{4}x > 1\end{equation}\].

- Isolate the variable

To isolate the variable x, multiply both sides of the inequality by \[\begin{equation}-4\end{equation}\](the reciprocal of \[\begin{equation}-\frac{1}{4}\end{equation}\]). Remember, when you multiply or divide an inequality by a negative number, you must reverse the inequality sign.

- Perform the multiplication and reverse the inequality

Multiply both sides by -4:\[\begin{equation}-4 \cdot (-\frac{1}{4}x) < 1 \cdot (-4)\end{equation}\].Simplifying this, we get\[\begin{equation}x < -4\end{equation}\].

- Write the final inequality

The final inequality statement is\[\begin{equation}x < -4\end{equation}\].

Key concepts

Isolating the variable

When dealing with inequalities, isolating the variable means getting the variable by itself on one side of the inequality.
This helps you see the solution more clearly.
Here's how to do it:

• Identify the variable you need to isolate.
• Use mathematical operations to move other numbers to the opposite side.
• Perform the same operation on both sides of the inequality to keep it balanced.

In our exercise, the given inequality is \( -\frac{1}{4}x > 1 \).
To isolate the variable x, we need to remove the \( -\frac{1}{4} \) coefficient.
This involves multiplication or division, which leads us to the next concept.

Multiplying or dividing inequalities

Multiplying or dividing inequalities by a number is a common step in isolating the variable.
However, there are special rules you need to follow, especially when dealing with negative numbers:

• If you multiply or divide both sides of an inequality by a positive number, the inequality direction stays the same.
• If you multiply or divide by a negative number, you must reverse the inequality sign.

In our exercise, we multiply both sides by \( -4 \), which is the reciprocal of \( -\frac{1}{4} \).
This will cancel out the negative fraction on the left side:

Performing the multiplication: \[ -4 \cdot ( -\frac{1}{4}x ) < 1 \cdot ( -4 ) \]
This simplifies to \( x < -4 \), but remember, we reversed the inequality sign because we multiplied by a negative number.

Reversing inequality signs

Reversing inequality signs is crucial when multiplying or dividing by a negative number.
It ensures the direction of the inequality remains correct.
Failing to reverse the inequality sign will give you the wrong solution.
Think of it this way: when you multiply or divide by a negative number, you flip the entire inequality.
In our example, when we multiply by \( -4 \), the inequality \( -\frac{1}{4}x > 1 \) becomes \( x < -4 \).
This reversal keeps the inequality true after the multiplication.
Remember, always check if you are dealing with a negative number to decide if you need to reverse the sign.