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Chapter 2: Problem 24

Solve each inequality and graph the solution on the number line. $$7 x+6 \geq 4 x$$

Short Answer

Expert verified
The solution to the inequality is \( x \geq -2 \). This is represented on the number line by a closed circle at -2 and shading to the right.

Step by step solution

01

Subtract \( 4x \) from both sides

By subtracting \( 4x \) from both sides, we obtain \( 3x + 6 \geq 0 \).
02

Subtract 6 from both sides

Subtract 6 from both sides to isolate \( 3x \) on one side of the inequality. This gives \( 3x \geq -6 \).
03

Divide by 3

Lastly, divide both sides of the inequality by 3 to isolate \( x \). This yields \( x \geq -2 \).
04

Graph the solution on a number line

The solution set is all real numbers greater than or equal to -2. On the number line, this is represented by a closed circle at -2 and shading to the right.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Number Line Graphing
Creating a graph on a number line involves depicting solutions visually to understand the range of possible answers. In the case of inequalities, this visualization is both helpful and simple. Each point on the number line represents a possible value of the variable.
For the inequality \( x \geq -2 \), the circle at \(-2\) is closed because \(-2\) itself is included in the solution (due to the greater than or equal to sign).
  • Closed circles signify inclusion (\( \geq \) or \( \leq \)).
  • Open circles indicate values not included (\( > \) or \( < \)).
  • A shaded line to the right or left shows the direction and range of the solution values.
This step-by-step approach makes it easier to see the values that satisfy the inequality.
Algebraic Expressions
An algebraic expression is a mathematical phrase combining numbers and variables using operations like addition, subtraction, multiplication, and division. In our inequality, we started with the expression \( 7x + 6 \geq 4x \). Understanding how these terms interact is key to solving the problem.
Here are important aspects to remember:
  • 7x and 4x are algebraic terms that represent unknowns multiplied by coefficients.
  • The expression operates under a basic principle: similar terms can be combined or eliminated.
  • Mathematical operations on algebraic expressions are the same as arithmetic ones 鈥� they follow rules of balancing or keeping both sides of an equation or inequality equal.
Breaking down complex expressions into simpler ones helps solve inequalities efficiently.
Isolation of Variable
The process of isolating a variable involves manipulating the inequality to express the variable alone on one side. This technique is crucial for understanding the solution to the inequality.
The steps of isolation in our example involve:
  • First, remove or combine like terms on both sides to bring the variable terms together. Here, we subtracted \( 4x \) from both sides: \( 3x + 6 \geq 0 \).
  • Next, eliminate any additional constants attached to the variable by performing inverse operations. Thus, subtract 6 to get \( 3x \geq -6 \).
  • Finally, divide by the coefficient of \( x \) to leave \( x \) isolated, resulting in \( x \geq -2 \).
By breaking down the steps, solving inequalities by isolating variables becomes clear and straightforward. Understanding and practicing this method will enhance your capability to solve similar mathematical problems.

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