Question

Fill in the blank to form a correct inequality statement. If \(x \geq 5,\) then \(-4 x______-20\)

Short answer

The correct inequality statement is \(-4x \leq -20\)

Step-by-step solution

Understanding the given inequality

We are given the inequality condition: \(x \geq 5\) and we need to determine the relationship between \(-4x\) and \(-20\).

Multiply the inequality by -4

Multiply both sides of the given inequality by -4. Remember that when we multiply an inequality by a negative number, the direction of the inequality sign changes: \( x \geq 5 \rightarrow -4x \leq -4 \times 5 \)

Simplify the right side

Simplify the right side of the inequality: \(-4x \leq -20\)

Key concepts

solving inequalities

Solving inequalities is a vital part of algebra and involves finding the values that satisfy a given inequality. Unlike equations, inequalities show a range of possible solutions rather than a single solution. For example, if we have an inequality like \(x \geq 5\), it means that \(x\) can be any number that is greater than or equal to 5.
To solve inequalities, we perform similar operations as we do with equations, such as addition, subtraction, multiplication, and division. However, there are some additional rules we need to follow especially when dealing with inequalities to ensure that the inequality sign remains correct.

multiplying inequalities

When multiplying inequalities, it is important to remember that multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality sign. For example, if we have \(x > 3\) and we multiply both sides by 2, the inequality remains the same: \(2x > 6\).
However, multiplying or dividing both sides of an inequality by a negative number does change the direction of the inequality sign. This is a crucial rule to remember. For instance, if we have \(x > 3\) and we multiply both sides by \(-2\), the inequality changes direction: \(-2x < -6\). This is because multiplying by a negative number essentially flips the number line, reversing the inequality.

inequality direction change

One of the key points in handling inequalities is understanding how and why the direction of the inequality sign changes. As seen in the original exercise, if we start with \(x \geq 5\) and we multiply both sides by -4, the inequality becomes \(-4x \leq -20\). This happens because multiplying by a negative number changes the order of values. For instance, -2 is less than -1, even though 2 is greater than 1.
Let’s break it down: If \(a > b\), and we multiply both sides by a positive number \(c\), then \(ac > bc\). But if \(a > b\) and we multiply by a negative number \(c\), then \(ac < bc\). The direction switch ensures that the inequality correctly represents the relationship between the two sides after the multiplication.