Question

Fill in the blank to form a correct inequality statement. If \(-\frac{1}{2} x \leq 3,\) then \(x____-6\)

Short answer

x \geq -6

Step-by-step solution

Understand the given inequality

The given inequality is \(-\frac{1}{2} x \leq 3\). We need to isolate the variable x to solve for it.

Multiply both sides by 2

To get rid of the fraction, multiply both sides of the inequality by -2. Remember that multiplying both sides of an inequality by a negative number reverses the inequality sign. \[ -2(-\frac{1}{2} x) \geq -2(3) \]

Simplify both sides

Simplify the expression to solve for x: \[ x \geq -6 \]

Key concepts

multiplying inequalities

When solving inequalities, sometimes you need to eliminate fractions or decimals. One common technique is multiplying to clear denominators. In our example, \(-\frac{1}{2} x \leq 3\), we see a fraction with a negative sign. To isolate x, we multiply both sides by -2.
Multiplying both sides of an inequality by a negative number introduces an important rule: the inequality sign must be reversed. So, begin multiplying both sides:

• Multiply the left side: \(-2(-\frac{1}{2} x)\)
• Multiply the right side: \(-2(3)\)

This multiplication step results in \(x \geq -6\), but note how we switched \(\leq \) to \(\geq \). Always remember to reverse the sign when multiplying by a negative.

reversing inequality sign

Reversing the inequality sign is a unique rule specific to inequalities involving multiplication or division by negative numbers. In our case, we started with \( -\frac{1}{2} x \leq 3\).
When we multiplied both sides by -2, we applied the rule of sign reversal. Why does this happen? Because multiplying or dividing by a negative flips the number line.

• Visualize: \(\frac{1}{2} \leq 2\)\footnote{Sample footer content}
• Multiply both sides by -1: \(-\frac{1}{2} \geq -2\)

Notice how the inequality sign changed direction. In our problem, after multiplying by -2, we reverse \( -\frac{1}{2} x \leq 3\) to \( x \geq -6\). This rule is crucial for ensuring the inequality remains true.

solving linear inequalities

Solving linear inequalities involves steps similar to those used for linear equations but requires careful attention to inequality rules. Let's break down the example:
The given inequality: \(-\frac{1}{2} x \leq 3\)

• First, identify the variable we need to isolate: x.
• Multiply both sides by -2 to handle the fraction: \(-2(-\frac{1}{2} x) \geq -2(3)\)

Simplify both sides:

• Left side: x
• Right side: -6

Result is: \( x \geq -6\)
Solving linear inequalities means not just performing algebraic operations, but also keeping inequalities balanced. Pay attention to special rules, especially when negative numbers come into play. This ensures you achieve a correct solution, as we did with \( x \geq -6\).