Question

Solve \(y0\). A. \(x<\frac{m}{y-b}\) B. \(x>\frac{m}{y-b}\) C. \(x>\frac{y-b}{m}\) D. \(x<\frac{y-b}{m}\)

Short answer

Option C: \( x > \frac{y - b}{m} \)

Step-by-step solution

Write the Given Inequality

Start with the inequality provided in the problem: \[ y < m x + b \]

Isolate the Linear Term

Subtract \( b \) from both sides to isolate the term with \( x \): \[ y - b < m x \]

Solve for \( x \)

Now, divide both sides by \( m \) to solve for \( x \): \[ \frac{y - b}{m} < x \]

Rewrite the Inequality

By changing the order for clarity, the inequality becomes: \[ x > \frac{y - b}{m} \]

Match with Given Options

Among the given multiple choices, find the one that corresponds to our solution: Option C: \( x > \frac{y - b}{m} \)

Key concepts

Linear Inequalities

Linear inequalities are mathematical expressions involving a linear function in which one side is not strictly equal to the other. Instead, it involves '<' (less than), '>' (greater than), '≤' (less than or equal to), or '≥' (greater than or equal to). For example, the inequality provided in the exercise is \( y < mx + b \). This means that the value represented on the left side of the inequality is less than the value represented on the right side. Linear inequalities are similar to linear equations but with these inequality signs instead of an equal sign.

Isolating Variables

Isolating a variable in an inequality involves rearranging the terms so that the variable of interest is on one side of the inequality sign and everything else is on the other side. In this exercise, the aim is to isolate the variable \(x\). To do this, follow these steps:

• First, subtract \(b\) from both sides to move the constant term to the left side: \[ y - b < mx \]
• Next, divide both sides by the coefficient of \(x\), which is \(m\). Since \(m\) is greater than zero, the inequality sign does not change: \[ \frac{y - b}{m} < x \]
• Finally, rewrite the inequality for clarity, making sure that the isolated variable \[ x \] is on the left side: \[ x > \frac{y - b}{m} \]

This process ensures that the inequality properly reflects the relationship between x and the other variables.

Solving for a Variable

When solving for a variable, especially in inequalities, ensure you perform each mathematical operation correctly to maintain the inequality's validity. Here, the goal was to solve for \(x\) while ensuring steps are followed carefully:

• First, isolate the linear term (the term with x) by moving all other terms to the opposite side of the inequality sign. This often means adding or subtracting terms from both sides.
• Second, if the variable is multiplied by a coefficient (such as m), divide every term by that coefficient to solve for the variable.
• Remember: If you multiply or divide by a negative number, the direction of the inequality sign changes. This does not apply here since \( m \) is greater than 0.

Practicing these steps in a step-by-step manner helps in understanding how to tackle similar problems in future exercises.

GED Test Math

The GED (General Educational Development) test includes a math section that often involves solving linear inequalities and other algebraic expressions. Understanding how to solve these types of problems efficiently is crucial:

• Read the problem carefully and identify what is asked.
• Translate the given word problems into mathematical inequalities or equations.
• Apply algebraic methods to isolate variables and solve the inequalities step-by-step, just as shown in the exercise provided.
• Double-check your steps to ensure the accuracy of your solution.
• Match your solution with the given answer choices if it is a multiple-choice question.

By mastering these concepts, you will be better prepared to handle a variety of problems on the GED math test and improve your overall score.