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Chapter 1: Problem 15

Linear Inequalities Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$2 x-5 > 3$$

Short Answer

Expert verified
Solution: \((4, \infty)\); open circle at 4, shade right.

Step by step solution

01

Isolate the Variable

To solve the inequality, start by isolating the variable term. The inequality given is \(2x - 5 > 3\). Add 5 to both sides to remove the constant from the left side, resulting in \(2x > 8\).
02

Solve for x

Next, divide both sides by 2 to solve for \(x\). This gives \(x > 4\).
03

Express in Interval Notation

The solution to the inequality \(x > 4\) can be expressed in interval notation as \((4, \infty)\).
04

Graph the Solution Set

To graph the solution set, draw a number line. Place an open circle at 4 (since 4 is not included) and shade the line to the right of 4 to indicate all numbers greater than 4.

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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 of writing subsets of the real numbers on a number line. It is especially useful when conveying the solution to inequalities like the one we're dealing with here. In essence, it uses a combination of numbers and round or square brackets to describe intervals.
For example, in the inequality solution, we have a simple statement:
  • The inequality is: \( x > 4 \).
  • This is represented in interval notation as: \((4, \infty)\).
In interval notation:
  • The number 4 is a boundary, meaning the interval starts just beyond it. We show this by using an open parenthesis, \(( \), indicating that 4 is not part of the solution.
  • We write \( \infty \) (infinity) to say the interval extends indefinitely in the positive direction.
  • Open intervals use \(( \text{or} )\) to show numbers not included in the set. Closed intervals use \([ \text{or} ]\), indicating numbers are included, e.g., \([a, b]\) includes both \(a\) and \(b\).
Getting the interval notation right means knowing whether endpoints are included (which they aren't here) or not. This provides a visual cue about the openness of the interval.
Solving Inequalities
Solving inequalities is somewhat like solving equations, but with an important difference: inequality signs can change direction. This is particularly true when multiplying or dividing by a negative number.
In our example, the inequality problem is given as:
  • Start with: \(2x - 5 > 3\)
  • First, isolate \(x\) by adding 5 to both sides: \(2x > 8\)
  • Divide both sides by 2: \(x > 4\)
While working through these steps, remember the following:
  • If you multiply or divide by a negative number, flip the inequality sign.
  • The goal is to isolate the variable on one side of the inequality.
  • It's often helpful to re-check your steps to ensure each transformation is valid under inequality rules.
Treat inequalities with care, especially when handling negative values. Understand the logical transformations, as these will guide you to the correct solution.
Graphing Solutions
Graphing solutions is a vital step to visualize the answer of an inequality. It involves plotting the solution on a number line to make the solution set clear. This pictorial representation helps in understanding exactly which numbers satisfy the inequality.
To graph the solution \(x > 4\) of our inequality:
  • Draw a straight horizontal line to represent a number line.
  • Mark the point 4 on this line. Because the inequality is "greater than" and not "equal to," use an open circle at 4.
  • Shade the line to the right of 4 to indicate all numbers larger than 4 are solutions.
This graphical representation does more than just relay the solution:
  • The open circle at 4 specifically shows that while our numbers get close to 4, 4 itself is not included.
  • The shaded part extends infinitely to the right, communicating that there is no upper bound to the values that \(x\) can take.
Graphing is not just a step but a method to intuitively understand inequalities. It provides clarity that can be especially helpful in multi-part problems or when discussing solutions with others.

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Most popular questions from this chapter

DISCOVER - PROVE: Relationship Between Solutions and Coefficients The Quadratic Formula gives us the solutions of a quadratic equation from its coefficients. We can also obtain the coefficients from the solutions. (a) Find the solutions of the equation \(x^{2}-9 x+20=0\) and show that the product of the solutions is the constant term 20 and the sum of the solutions is \(9,\) the negative of the coefficient of \(x\) (b) Show that the same relationship between solutions and coefficients holds for the following equations:$$ \begin{array}{l}x^{2}-2 x-8=0 \\\x^{2}+4 x+2=0\end{array}$$ (c) Use the Quadratic Formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has solutions \(r_{1}\) and \(r_{2}\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)

Simplify the expression. (a) \(5^{2 / 3} \cdot 5^{1 / 3}\) (b) \(\frac{3^{3 / 5}}{3^{2 / 5}}\) (c) \((\sqrt[3]{4})^{3}\)

Suppose an object is dropped from a height \(h_{0}\) above the ground. Then its height after \(t\) seconds is given by \(h=-16 t^{2}+h_{0},\) where \(h\) is measured in feet. Use this information to solve the problem. If a ball is dropped from 288 ft above the ground, how long does it take to reach ground level?

Write the number indicated in each statement in scientific notation. (a) A light-year, the distance that light travels in one year, is about \(5,900,000,000,000 \mathrm{mi}\) (b) The diameter of an electron is about \(0.0000000000004 \mathrm{cm}\) (c) A drop of water contains more than 33 billion billion molecules.

Shrinkage in Concrete Beams As concrete dries, it shrinks - the higher the water content, the greater the shrinkage. If a concrete beam has a water content of \(\bar{w} \mathrm{kg} / \mathrm{m}^{3},\) then it will shrink by a factor $$S=\frac{0.032 w-2.5}{10,000}$$ where \(S\) is the fraction of the original beam length that disappears due to shrinkage. (a) A beam \(12.025 \mathrm{m}\) long is cast in concrete that contains \(250 \mathrm{kg} / \mathrm{m}^{3}\) water. What is the shrinkage factor \(S ?\) How Iong will the beam be when it has dried? (b) A beam is \(10.014 \mathrm{m}\) long when wet. We want it to shrink to \(10.009 \mathrm{m},\) so the shrinkage factor should be \(S=0.00050 .\) What water content will provide this amount of shrinkage? PICTURE CANT COPY
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