Question

Solve the inequality. \(-18.2 x \geq-91\)

Short answer

The solution to the inequality \(-18.2 x \geq-91\) is \(x \leq 5\).

Step-by-step solution

Set the inequality

The inequality to solve is \(-18.2 x \geq-91\).

Isolate the variable \(x\)

Isolate \(x\) by dividing both sides of the inequality by -18.2. Remember that dividing or multiplying by a negative number changes the direction of the inequality sign. It changes from 'greater than or equal to' to 'less than or equal to'. The inequality thus becomes: \(x \leq \frac{-91}{-18.2}\).

Simplify the inequality

Simplify the inequality to get the solution. The inequality becomes \(x \leq 5\).

Key concepts

Inequality Manipulation

Mastering inequality manipulation is essential in solving mathematical inequalities efficiently and correctly. Imagine an inequality as a balance scale; when you alter one side, a corresponding change must be made on the other to maintain the balance. When faced with an inequality such as \( -18.2x \geq -91 \), one begins to solve it through a series of balanced maneuvers. Here, the goal is to isolate the variable on one side, similarly to how one would approach an equation.

To keep our scale balanced, we manipulate the inequality by performing the same operation on both sides. In this case, since we want to isolate \(x\), we decide to divide both sides by \( -18.2 \), the coefficient of \(x\). It’s essential to treat the inequality symbol with care during manipulation. Unlike equations, certain operations, such as multiplying or dividing by a negative number, require us to flip the inequality sign. This ensures that the relational integrity of the inequality is preserved.

Isolation of Variables

Isolation of variables is like finding the main character in a story. In math, isolating the variable means to rewrite the inequality, so the variable you're solving for stands alone on one side of the inequality symbol. Think of it as clearing the stage so that the variable can shine in the spotlight without any distractions.

In the given inequality \( -18.2x \geq -91 \), our main character is \(x\). To isolate \(x\), we need to eliminate all other numbers from its side. We achieve this by dividing both sides of the inequality by \( -18.2 \) because \( -18.2 \) is currently multiplied by \(x\). Once we've done this division, and thus 'freed' \(x\) from other terms, we're left with \(x \leq 5\). This is our variable shining on its own, telling us that \(x\) can be any number less than or equal to 5.

Inequality Reversal

The inequality reversal concept takes center stage when multiplication or division by a negative number is involved in an inequality. This concept might feel counterintuitive, but it's a fundamental rule within the realm of inequalities. Unlike equations, where an equal sign remains unbothered by such operations, the way we read inequalities must adapt to these actions.

To understand inequality reversal, remember this simple rule: Whenever you multiply or divide both sides of an inequality by a negative number, flip the inequality sign. For instance, with \( -18.2x \geq -91 \), dividing by \( -18.2 \) (a negative) flips the sign from 'greater than or equal to' to 'less than or equal to'. This critical move is what ensures the solution \(x \leq 5\) correctly represents all the possible values of \(x\) that satisfy the original inequality.