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Chapter 1: Problem 60

Solve the inequality. Express your answer in interval notation. $$-2 x-1 \geq \frac{x+5}{2}$$

Short Answer

Expert verified
The solution for the inequality is \( x \leq -\frac{7}{5} \), or in interval notation, \( (-\infty, -\frac{7}{5}] \).

Step by step solution

01

Isolate x

Begin by multiplying every term by 2 to eliminate the fraction:\[-4x - 2 \geq x + 5.\]After that, bring all x terms to one side and constants to the other:\[ -4x - x \geq 5 + 2,\]This simplifies to:\[-5x \geq 7.\]
02

Solve for x

To solve for x, divide each side by -5. Whenever we divide or multiply by a negative number in an inequality, we must remember to flip the direction of the inequality. Thus,\[ x \leq -\frac{7}{5}.\]
03

Express solution in interval notation

The solution in interval notation is: \(x\) is within the interval from \(-\infty\) to \(-\frac{7}{5}\), inclusive on \(-\frac{7}{5}\). This is represented as \( (-\infty, -\frac{7}{5}] \).

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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 that make an inequality true. This method uses brackets and parentheses.
Brackets, like \([a, b]\), mean the endpoint is included, and parentheses, like \((a, b)\), mean the endpoint is not included.

For example:
  • \([3, 7]\) includes all numbers from 3 to 7, and both 3 and 7 are part of the set.
  • \((3, 7]\) includes numbers greater than 3 and up to 7, but not 3 itself.
  • \((-\infty, -\frac{7}{5}]\) includes all numbers up to and including \(-\frac{7}{5}\), but any negative infinity point is always open (\(-\infty\)), as it does not represent an actual number.
Using interval notation makes it easy to see which numbers are solutions to an inequality.
Solving Inequalities
Solving inequalities is similar to solving regular equations, but there is a key difference involving the inequality sign.
When you solve inequalities, you aim to isolate the variable on one side, just like in equations.

For example, let's solve the inequality \(-2x - 1 \geq \frac{x + 5}{2}\):
  • First, eliminate fractions by multiplying all terms by 2, giving us \(-4x - 2 \geq x + 5\).
  • Next, get all terms involving \(x\) on one side: \(-4x - x \geq 5 + 2\), simplifying to \(-5x \geq 7\).
  • Finally, solve for \(x\) by dividing by \(-5\), remembering to flip the inequality to \(x \leq -\frac{7}{5}\).
Always focus on keeping the inequality balanced as you manipulate it.
Multiplying by Negative Numbers in Inequalities
When we multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be flipped.
This rule is crucial to remember, as it keeps the inequality logically consistent.

Here's why:
  • Consider the numbers \(-3\) and \(2\). It's clear that \(-3 < 2\).
  • If we multiply both by \(-1\), we get \(3 > -2\), so the direction is flipped.
  • This same rule applied when we reached \(-5x \geq 7\) in our example. Dividing by \(-5\) gave us \(x \leq -\frac{7}{5}\).
Flipping the inequality ensures our solution remains correct and reflects the true relationship between the numbers.

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Most popular questions from this chapter

Let \(f(s)=m s+b .\) Find values of \(m\) and \(b\) such that \(f(0)=2\) and \(f(2)=-4 .\) Write an expression for the linear function \(f(s) .\) (Hint: Start by using the given information to write down the coordinates of two points that satisfy \(f(s)=m s+b .)\)

Travel This problem is an extension of Example \(1 .\) A one-way ticket on a weckday from Newark, New Jersey, to New York, New York, costs 3.30 dollars for a train departing during peak hours and 2.50 dollars for a train departing during off-peak hours. Peak morning hours are from 6 A.M. to 10 A.M. and peak evening hours are from 4 P.M. to 7 P.M. The rest of the day is considered to be off-peak. (Source: New Jersey Transit) (a) Construct a table that takes the time of day as its input and gives the fare as its output. (b) Write the fare as a function of the time of day using piecewise function notation. (c) Graph the function.

A jogger on a pre-set treadmill burns 3.2 calories per minute. How long must she jog to burn at least 200 calories?

Solve the inequality. Express your answer in interval notation. $$\frac{1}{3}(x+1)

This set of exercises will draw on the ideas presented in this section and your general math background. Let \(f\) be defined as followos. $$ f(x)=\left\\{\begin{array}{ll} 0, & \text { if } x \leq 1 \\ 2, & \text { if } x>1 \end{array}\right. $$ Graph \(f(x-1)\)
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