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Magazine Chapter 2: Problem 26
Solve each inequality. Graph the solution set and write it in interval notation. $$ 3(5 x-4) \leq 4(3 x-2) $$
Short Answer
Expert verified
The solution is \(x \leq \frac{4}{3}\) or \((-
\infty, \frac{4}{3}]\).
Step by step solution
01
Distribute Constants
Begin by distributing the constants outside of the parentheses for both sides of the inequality.On the left side: \[3(5x - 4) = 15x - 12\]On the right side:\[4(3x - 2) = 12x - 8\]
02
Set Up the Inequality
After distribution, rewrite the inequality with the new expressions.\[15x - 12 \leq 12x - 8\]
03
Rearrange Terms
Move all terms involving \(x\) to one side and constant terms to the other side. Subtract \(12x\) from both sides of the inequality.\[15x - 12x - 12 \leq -8\]This simplifies to:\[3x - 12 \leq -8\]
04
Isolate the Variable Term
Add 12 to both sides of the inequality to isolate the \(x\) term.\[3x \leq 4\]
05
Solve for x
Divide both sides by 3 to solve for \(x\).\[x \leq \frac{4}{3}\]
06
Write in Interval Notation
The solution \(x \leq \frac{4}{3}\) in interval notation is written as:\((- \infty, \frac{4}{3}]\).
07
Graph the Solution Set
To graph \(x \leq \frac{4}{3}\), draw a number line, shade the region to the left of \(\frac{4}{3}\) and include \(\frac{4}{3}\) with a closed circle. This indicates that all numbers less than or equal to \(\frac{4}{3}\) are part of the solution set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a way to describe a set of numbers between a pair of endpoints on a number line. When we solve inequalities, we often use interval notation to express the solution set in shorthand. For example, the solution to the inequality \( x \leq \frac{4}{3} \) can be written as \( (-\infty, \frac{4}{3}] \) in interval notation.
Here's what it means:
Here's what it means:
- The parenthesis \( (-\infty, ...) \) indicates that the interval extends indefinitely to the left, capturing all smaller numbers.
- The closed bracket \( ... ] \) at \( \frac{4}{3} \) means that this endpoint is included in the interval.
Graphing Inequalities
Graphing inequalities on a number line is a visual method to understand the solutions of inequalities, and it often complements interval notation. When graphing an inequality like \( x \leq \frac{4}{3} \), you'll need to show graphically which values of \( x \) satisfy the condition.
Steps to Graph:
Steps to Graph:
- Draw a horizontal line representing the number line, and mark the number \( \frac{4}{3} \) on it.
- Use a closed circle at \( \frac{4}{3} \) if the inequality includes equal to ("\( \leq \)" or "\( \geq \)").
- Shade the area to the left of \( \frac{4}{3} \) to indicate that all numbers less than \( \frac{4}{3} \) are included.
Algebraic Expressions
An algebraic expression consists of numbers, variables, and operators combined to represent a value. When solving inequalities, understanding how to manipulate these expressions is crucial.
Key Points:
Key Points:
- Distributing Constants: Multiply every term inside a parenthesis by the constant outside, as shown when transitioning \( 3(5x - 4) \) into \( 15x - 12 \).
- Combining Like Terms: Group terms that contain the same variables, such as collecting all \( x \)-terms on one side of the inequality.
- Isolating the Variable: Use algebraic operations like addition, subtraction, multiplication, or division to solve for the variable, providing steps to reach the inequality solution \( x \leq \frac{4}{3} \).
