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Chapter 2: Problem 3

Solve the inequality and specify the answer using interval notation. $$4 x+6 < 3(x-1)-x$$

Short Answer

Expert verified
The solution is \((-\infty, -\frac{9}{2})\) in interval notation.

Step by step solution

01

Simplify the Right Side

First, let's simplify the right side of the inequality equation \(3(x-1) - x\). Distribute the 3 over \(x-1\) to get: \[ 3(x-1) = 3x - 3 \]This simplifies the inequality to:\[ 4x + 6 < 3x - 3 - x \]
02

Combine Like Terms on the Right Side

Now, combine the terms \(3x - x\) on the right side of the inequality. This gives us:\[ 4x + 6 < 2x - 3 \]
03

Move x Terms to One Side

Subtract \(2x\) from both sides to bring all the \(x\) terms to one side of the inequality:\[ 4x - 2x + 6 < 2x - 2x - 3 \]This leads to:\[ 2x + 6 < -3 \]
04

Isolate the Term with x

Subtract 6 from both sides to isolate the term with \(x\):\[ 2x + 6 - 6 < -3 - 6 \]This simplifies to:\[ 2x < -9 \]
05

Solve for x

Finally, divide both sides by 2 to solve for \(x\):\[ \frac{2x}{2} < \frac{-9}{2} \]Thus, we have:\[ x < -\frac{9}{2} \]
06

Write in Interval Notation

To express the solution in interval notation, we recognize that \(x\) is less than \(-\frac{9}{2}\). In interval notation, this is written as:\[ (-\infty, -\frac{9}{2}) \]

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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 shorthand used to describe a range of values. It's often used in inequalities to show where a variable lies on the number line.
When you see something like \[ (-\infty, -\frac{9}{2}) \] this means that the variable can take on any value less than \(-\frac{9}{2}\), but never actually equals \(-\frac{9}{2}\).
  • "(" or ")" are used when the endpoint is not included, meaning it is open.
  • "[" or "]" mean the endpoint is included, making it closed.
Infinity \((-\infty \text{ or } + \infty)\) is always expressed with open intervals because they are not specific values.
Distributive Property
The distributive property is a fundamental principle used to remove parentheses and distribute a factor across terms inside the parentheses.
If you have an expression like \(3(x-1)\), applying the distributive property gives you \(3 \cdot x - 3 \cdot 1\), resulting in \(3x - 3\).
This step is crucial because it simplifies complex expressions by breaking them down into more helpful parts.
It鈥檚 especially useful in inequalities and equations where you need to ensure every term is handled correctly.
Remember:
  • Multiply what is outside the parentheses with each term inside.
  • Respect the distribution of negative signs if present.
Combining Like Terms
Combining like terms means reducing expressions by adding or subtracting terms that are the same type.
In our inequality, we combined \(3x\text{ and }(-x)\) from \(3x - x\) to get \(2x\).
This simplification helps maintain clarity and eases solving, as you are dealing with fewer terms.
  • Terms that are "alike" have identical variable parts and exponents.
  • Only coefficients (the numbers multiplying the variables) are combined.
By streamlining expressions, it鈥檚 easier to manipulate and solve them.
Isolation of Variables
Isolating the variable means getting the variable you're solving for (usually \(x\)) on one side of the equation or inequality.
This process ensures that you have exactly what \(x\) equals or is less than, in terms of another value or expression.
For instance, when we moved \(2x\) from the right to the left by subtracting \(2x\) from both sides, we were isolating \(x\).
To completely isolate \(x\), follow these steps:
  • Move terms without the variable to the opposite side by adding or subtracting them.
  • Divide or multiply both sides to solve for \(x\) completely.
Isolating helps to clearly see what the variable represents in the inequality.

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Most popular questions from this chapter

In the United States over the years \(1980-2000\), sulfur dioxide emissions due to the burning of fossil fuels can be approximated by the equation $$y=-0.4743 t+24.086$$ where \(y\) represents the sulfur dioxide emissions (in millions of tons) for the year \(t\), with \(t=0\) corresponding to \(1980 .\) Source: This equation (and the equation in Exercise 48) were computed using data from the book Vital Signs 1999 Lester Brown et al. (New York: W. W. Norton \& \(\mathrm{Co} ., 1999\) ). (a) Use a graphing utility to graph the equation \(y=-0.4743 t+24.086\) in the viewing rectangle [0,25,5] by \([0,30,5] .\) According to the graph, sulfur dioxide emissions are decreasing. What piece of information in the equation \(y=-0.4743 t+24.086\) tells you this even before looking at the graph? (b) Assuming this equation remains valid, estimate the year in which sulfur dioxide emissions in the United States might fall below 10 million tons per year. (You need to solve the inequality \(-0.4743 t+24.086 \leq 10 .)\)

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Solve the inequalities Suggestion: A calculator may be useful for approximating key numbers. $$(x-3)^{2}(x+1)^{4}(2 x+1)^{4}(3 x+2) \leq 0$$

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