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Chapter 2: Problem 24

Solve inequality. Write the solution set interval notation, and graph it. \(5 x-2<4 x-5\)

Short Answer

Expert verified
The solution is \(x < -3\), which in interval notation is \((-\infty, -3)\).

Step by step solution

01

Combine like terms

First, subtract \(4x\) from both sides of the inequality to isolate the \(x\) term on one side: \(5x - 2 - 4x < 4x - 4x - 5\). This simplifies to \(x - 2 < -5\).
02

Isolate x

Next, add 2 to both sides to isolate the \(x\) term completely: \(x - 2 + 2 < -5 + 2\). This simplifies to \(x < -3\).
03

Write in interval notation

The solution \(x < -3\) can be expressed in interval notation as \((-\infty, -3)\).
04

Graph the solution set

To graph \(x < -3\), draw a number line, plot an open circle at \(-3\) to indicate that \(-3\) is not included in the solution set, and shade all the numbers to the left of \(-3\).

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Key Concepts

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

Solving Inequalities
Solving inequalities is an essential skill in algebra that helps us find the range of values that satisfy a given condition. In this exercise, we are asked to solve the inequality: \(5x - 2 < 4x - 5\)The process involves several steps:

First, we combine like terms. We subtract \(4x\) from both sides of the inequality to isolate the \(x\) term on one side:\(5x - 2 - 4x < 4x - 4x - 5\).This simplifies to:\(x - 2 < -5\).Next, we isolate \(x\) by adding 2 to both sides:\(x - 2 + 2 < -5 + 2\),which simplifies further to:\(x < -3\).We have found that any value of \(x\) less than \(-3\) will satisfy the inequality.
Interval Notation
Once the inequality \(x < -3\) has been solved, we can express the solution in interval notation. Interval notation provides a clean and concise way to represent a range of values. When using interval notation, remember:
  • Parentheses \(( )\) indicate that an endpoint is not included in the interval.
  • Square brackets \([ ]\) indicate that an endpoint is included in the interval.
For our solution, since \(x\) is less than \(-3\) and \(-3\) itself is not included, we use a parenthesis:\((-\infty, -3)\).This indicates that the solution set includes all numbers less than \(-3\) but not \(-3\) itself.
Graphing Inequalities
Visualizing the solution through graphing helps to better understand the range of values that satisfy the inequality. To graph the inequality \(x < -3\):
  • Draw a number line.
  • Mark the point \(-3\) on the number line.
  • Draw an open circle at \(-3\) to indicate that \(-3\) is not included in the solution set.
  • Shade the region to the left of \(-3\) to show all numbers less than \(-3\) are part of the solution.
This graphical representation confirms that our solution set includes all values less than \(-3\), matching our interval notation of \((-\infty, -3)\).

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