Problem 40 - Algebra 1

Question

SOLVING INEQUALITIES Solve the inequality. $$-12.3 x>86.1$$

Step-by-Step Solution

Answer

\( x < -7 \)

Step 1: Isolate x

To isolate x, divide both sides of the inequality by -12.3. However, when an inequality is divided by a negative number, the direction of the inequality changes. Here, the inequality \( -12.3x > 86.1 \) becomes \( x < -86.1 / -12.3 \).

Step 2: Calculate x

Simplify the right side by performing the division: \( x < -7 \).

Key Concepts

Inequality Manipulation

When solving inequalities, a key concept is the ability to 'manipulate' the inequality in such a way that you maintain its truth. Manipulating an inequality usually involves performing operations such as addition, subtraction, multiplication, or division to both sides of the inequality without changing the inequality’s sense.

Similar to equations, the goal is typically to isolate the variable on one side. However, unlike equations, the inequality symbol must be carefully observed. A particularly important rule is that when you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be flipped. For instance, if you have an inequality like \( -2y > 6 \), dividing both sides by -2 (a negative number) would reverse the inequality to \( y < -3 \) because this ensures the inequality continues to express the correct relationship between the two sides.

Always remember that the end goal is to clearly identify the set of numbers for which the inequality holds true, which may sometimes involve manipulating the inequality to a more simplified or standard form.

Division Property of Inequality

Flipping the Inequality

When working with the division property of inequality, it's crucial to remember that the direction of the inequality changes when dividing by a negative number. This is a stark contrast to equations, where you can divide by negative numbers freely without altering the direction of the equation.

Why Flip?

The reason behind flipping the inequality is rooted in the number line structure: if a negative value is divided by a negative number, the result is positive, and vice versa. This reversal keeps the inequality true. Consider this: if \( a > b \) and you divide both sides by \( -c \), where \( c > 0 \), you unexpectedly get a pair of opposites: \( -a < -b \) which means if you then divide by another negative, \( -1 \) for example, the resulting inequality must reverse to \( a/c < b/c \) to accurately reflect the original relationship.

Isolating the Variable

Isolation of the variable is the process of manipulating an inequality or equation so that the variable appears by itself on one side of the inequality symbol, and a number or an expression stands on the other side. This is performed through a series of operations that maintain the equality or inequality.

Operations to Isolate

To isolate the variable, you can use standard operations of addition, subtraction, multiplication, or division, but it's important to apply them carefully to both sides. In our original example of solving the inequality \( -12.3x > 86.1 \), we needed to divide by -12.3, which required us to flip the inequality. The resulting isolation \( x < -7 \) helps us understand that the solution to the inequality is all the numbers less than -7. The isolated variable clearly shows the boundary of the solution set and makes the inequality easier to visualize and understand.