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Chapter 1: Problem 53

Solve the inequality. Express your answer in interval notation. $$2-2 x \geq x-1$$

Short Answer

Expert verified
The solution to the inequality is \(x \leq 1\). In interval notation, this is express as \((-鈭�, 1]\)

Step by step solution

01

Combining like terms

Start by combining like terms on both sides of the inequality. This is done by adding \(2x\) to both sides of the equation to cancel out \(-2x\) from the left and add \(2x\) to \(x\) on the right. This results in \(2 \geq 3x - 1\)
02

Isolate the variable

Next, isolate \(3x\) on one side of the equation by adding 1 to both sides of the inequality to cancel out \(-1\) on the the right side. This results in \(3 \geq 3x\).
03

Solve the inequality

Finally, to solve the inequality for \(x\), divide both sides by 3. This results in \(x \leq 1\)

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Key Concepts

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

Interval Notation
When we solve inequalities, we often use interval notation to express the solutions. This method helps us clearly show the range of values that satisfy the inequality. For example, if the solution to an inequality is that a variable, say \(x\), can be any number less than or equal to 1, you would write this in interval notation as \((\infty, 1]\). Here鈥檚 what each part means:

  • The parenthesis \((\) indicates that infinity is never included. Infinity or negative infinity is always accompanied by a parenthesis because they are concepts, not numbers you can reach.
  • The square bracket \([]\) means that the number is included in the set. So, \([1]\) means \(x\) can be exactly 1.
Using interval notation makes reading and understanding solutions quicker and more standardized.
Inequalities
Inequalities are statements that show the relationship between two expressions that are not equal. They use symbols like \(>\), \(<\), \(\leq\), and \(\geq\). In the exercise, you dealt with \(\geq\), meaning 'greater than or equal to'.

The process of solving inequalities is much like solving equations, but with some added rules:
  • When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. This is crucial to remember to ensure your solution is correct.
  • Always simplify the inequality by combining like terms, just like you would in equations.
Understanding these rules helps avoid common mistakes and correctly interpret the inequality鈥檚 meaning.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to isolate a specific variable. In solving inequalities like the one in the exercise, these steps are essential:

  • Combining like terms: Bring similar terms together on each side of the inequality. For example, adding \(2x\) to both sides to simplify.
  • Isolating the variable: Move all terms containing the variable to one side and constants to the other. This often involves adding or subtracting values on both sides.
  • Solving for the variable: Divide or multiply to get the variable by itself. In this case, dividing by 3 gave us the result \(x \leq 1\).
These steps systematically simplify the inequality, making it easier to read and solve. This approach builds familiarity and confidence in tackling different types of algebraic problems.

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Most popular questions from this chapter

A long-distance telephone company advertises that it charges 1.00 dollars for the first 20 minutes of phone use and 7 cents a minute for every minute beyond the first 20 minutes. Let \(C(t)\) denote the total cost of a telephone call lasting \(t\) minutes. Assume that the minutes are nonnegative integers. (a) Many people will assume that it will cost only 0.50 dollars to talk for 10 minutes. Why is this incorrect? (b) Write an expression for the function \(C(t).\) (c) How much will it cost to talk for 5 minutes? 20 minutes? 30 minutes?

Car Ownership Costs In this problem, you will investigate whether it is cost effective to purchase a car that gets better gasoline mileage, even though its purchase price may be higher. A 2003 Subaru Outback wagon costs \(\$ 23,500\) and gets 22 miles per gallon. A 2003 Volkswagen Passat wagon costs \(\$ 24,110\) and gets 25 miles per gallon. Assume that gasoline costs \(\$ 4\) per gallon. (Sources: Edmunds.com and U.S. Environmental Protection Agency) (a) What is the cost of gasoline per mile for the Outback wagon? the Passat wagon? (b) Assume that the total cost of owning a car consists of the price of the car and the cost of gasoline. For each car, find a linear function describing the total cost, with the input variable being the number of miles driven. (c) What is the slope of the graph of each function in part (b), and what do the slopes signify? (d) How many miles would you have to drive for the total cost of the Passat to be the same as that of the Outback?

This set of exercises will draw on the ideas presented in this section and your general math background. Sketch the graph of \(f(x)=-3 x+2\) by hand. Use it to graph \(g(x)=|f(x)| .\) What is the \(x\) -intercept of the graph of \(g(x) ?\)

Solve the inequality. Express your answer in interval notation. $$0<-x+5<4$$

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