Problem 39 - Algebra 1

Question

SOLVING INEQUALITIES Solve the inequality. $$-\frac{x}{3} \geq 15$$

Step-by-Step Solution

Answer

So, the solution to the inequality \(-\frac{x}{3} \geq 15\) is \(x \leq -45\).

Step 1: Multiply both sides of the inequality by -3

In this step, both sides of the inequality will be multiplied by -3. Keep in mind that when you multiply or divide an inequality by a negative number, the direction of inequality changes. So, \(-\frac{x}{3} \geq 15\) becomes \(x \leq -45\).

Step 2: Check the solution

In order to be confidently sure about the correctness of the solution, substituting any number less than or equal to -45 to \(x\) in the original inequality would satisfy it.

Key Concepts

Inequality Properties

Inequalities are expressions that show the relationship between two values when they are not equal. An important aspect to remember when working with inequalities is that the inequality sign can change direction under certain conditions, which is not something that happens with equalities. Here are some key properties:

• Transitive Property: If \(a > b\) and \(b > c\), then \(a > c\). This helps in understanding how inequalities can be linked through intermediate values.
• Addition and Subtraction: Adding or subtracting the same number from both sides of an inequality doesn’t affect the inequality sign. For example, if \(a > b\), then \(a + c > b + c\).
• Multiplication and Division with Positives: When you multiply or divide both sides by a positive number, the direction of the inequality remains the same.
• Direction Change with Negatives: When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses. For example, if you multiply \(a < b\) by -1, it becomes \(a > b\).

Understanding these properties is crucial because they guide the steps you take to solve inequalities effectively.

Multiplying by Negative Numbers

When solving inequalities and you multiply or divide by a negative number, you must reverse the inequality sign. This rule is vital for ensuring the inequality remains true.

For example, in the original exercise \(-\frac{x}{3} \geq 15\), we multiplied both sides by \(-3\) to solve for \(x\). This multiplication changed the inequality from "greater than or equal to" to "less than or equal to". This is because multiplying by a negative number flips the direction:

• If \(a > b\), then multiplying both sides by \(-1\) gives \(a < b\).
• If \(c \leq d\), then multiplying both sides by \(-1\) results in \(c \geq d\).

Remember, this flipping rule only applies when multiplying or dividing by a negative, not for positive numbers or zero.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They can represent a wide range of mathematical scenarios.

In inequalities, algebraic expressions are often manipulated to isolate the variable of interest. Consider the problem \(-\frac{x}{3} \geq 15\). The fraction here can be thought of as the algebraic expression that needs to be simplified.

• The goal in solving these types of inequalities is to get the variable (in this case, \(x\)) by itself on one side of the inequality sign.
• Simplify the expression step by step: by multiplying or dividing terms to isolate the variable.
• Adjusting the inequality throughout the process by adhering to properties like those discussed, such as changing the direction of the inequality when necessary.

Understanding how to manipulate algebraic expressions helps not only in solving inequalities but also in unraveling more complex mathematical questions.