Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 16
Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$\frac{2}{5} x+1<\frac{1}{5}-2 x$$
Short Answer
Expert verified
The solution is \((-\infty, -\frac{1}{3})\).
Step by step solution
01
Clear Fractions by Multiplying by the Denominator
Multiply the entire inequality by 5 to eliminate the fractions. This leads to:\[ 5 \left( \frac{2}{5} x + 1 \right) < 5 \left( \frac{1}{5} - 2x \right) \]This simplifies to:\[ 2x + 5 < 1 - 10x \]
02
Collect Like Terms
Add \(10x\) to both sides of the inequality to collect the \(x\)-terms on one side:\[ 2x + 10x + 5 < 1 \]This simplifies to:\[ 12x + 5 < 1 \]
03
Isolate the Variable Term
Subtract 5 from both sides to isolate the term with \(x\):\[ 12x < 1 - 5 \]This simplifies to:\[ 12x < -4 \]
04
Solve for the Variable
Divide both sides by 12 to solve for \(x\):\[ x < \frac{-4}{12} \]Simplify the fraction:\[ x < -\frac{1}{3} \]
05
Express Solution in Interval Notation
The solution can be expressed in interval notation as:\[ (-\infty, -\frac{1}{3}) \]
06
Graph the Solution Set
To graph the solution, draw a number line. Mark \(-\frac{1}{3}\) with an open circle to indicate that it is not included in the solution set. Shade the line to the left of \(-\frac{1}{3}\) to represent all numbers less than \(-\frac{1}{3}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Linear Inequalities
Solving linear inequalities involves finding the range of values for a variable that satisfies the inequality condition. Unlike linear equations, inequalities use symbols such as <, >, 鈮�, or 鈮� instead of an equal sign. Let's take the inequality from the exercise: \[ \frac{2}{5}x + 1 < \frac{1}{5} - 2x \] Here is what you do step-by-step:
- Clear any fractions by multiplying the entire inequality by a common multiple. In the exercise, multiplying by 5 simplifies things nicely.
- Collect like terms to simplify the inequality. Gather variables on one side and constants on the other side.
- Isolate the variable by performing operations to remove constant terms on its side, such as addition or subtraction.
- Solve for the variable like you would in an equation, but remember that if you multiply or divide by a negative number, the inequality sign flips direction.
Interval Notation
Interval notation is a helpful way to describe a set of numbers between two endpoints. In the context of inequalities, it expresses all the solutions compactly and clearly.For the inequality \( x < -\frac{1}{3} \), the solution is all the numbers less than \(-\frac{1}{3}\). Using interval notation, we write this as:\[ (-\infty, -\frac{1}{3}) \] Here's why:
- Parens or Brackets? Use parentheses, \(( )\), to indicate that an endpoint is not included in the solution. Brackets, \([ ]\), would mean it's included. Here, because \(-\frac{1}{3}\) itself is not part of the solution, we use a parenthesis.
- Infinity Signs: Since numbers go on forever, \( -\infty \) represents the infinitely small numbers lower than your smallest bound, with an always-open parenthesis because infinity is not a number you can actually reach.
Graphing Inequalities
Graphing inequalities is a visual method to represent the solution set on a number line. This helps to understand which parts of the line are included in the solution.Here's how to graph \( x < -\frac{1}{3} \):
- Draw a horizontal line to represent the real number line.
- Locate \(-\frac{1}{3}\) on the line. This is where the inequality "cuts off."
- Put an open circle on \(-\frac{1}{3}\) to signify that it's not included in the solution (due to the 'less than' nature of the inequality).
- Shade the line extending to the left of \(-\frac{1}{3}\) to show that all numbers to the left are solutions.
