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Chapter 2: Problem 5

Are the inequalities \(x-4<5\) and \(x>9\) equivalent?

Short Answer

Expert verified
No, the inequalities \(x-4<5\) and \(x>9\) are not equivalent.

Step by step solution

01

Solve the first inequality

To solve the inequality \(x-4<5\), add 4 to both sides. The solution is \(x<9\). The solution set includes all real numbers less than 9.
02

Check the second inequality

The second inequality given is \(x>9\). The solution to this inequality is all real numbers greater than 9.
03

Compare the solution sets

As the solution sets for the two inequalities are completely different (one is less than 9 and the other is greater than 9), it can be concluded that the inequalities \(x-4<5\) and \(x>9\) are not equivalent.

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Key Concepts

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

Solving Inequalities
Solving inequalities involves finding all values of a variable that make the inequality true. Unlike equations, which have specific solutions, inequalities describe ranges of values. Let's consider the inequality \( x - 4 < 5 \).To solve it:
  • First, we isolate the variable \( x \) by performing operations that maintain the inequality. Here, we add 4 to both sides.
  • This results in \( x < 9 \).
The process involves simple arithmetic operations, such as addition, subtraction, multiplication, or division, while respecting the rules of inequality.
Remember to reverse the inequality sign when you multiply or divide both sides by a negative number. This is crucial to getting the correct solution.
Solution Set
The solution set of an inequality consists of all values that satisfy the inequality. For the inequality \( x < 9 \), the solution set includes:
  • All real numbers less than 9.
When we say "all real numbers less than 9," we imagine starting from negative infinity and moving up towards but not reaching 9.
It's like a number line that extends infinitely to the left.
The form of the solution set is often expressed using interval notation: \[ (-\infty, 9) \]. For the inequality \( x > 9 \), the solution set includes all real numbers greater than 9. It is denoted as:
  • \((9, +\infty)\), which shows that the actual number 9 is not included, but all numbers greater than 9 are part of the set.
Understanding solution sets helps visualize the span of solutions for inequalities.
Equivalent Inequalities
Equivalent inequalities are different inequalities that have the same solution set. In simpler terms, no matter how you write them, they represent the same range of solutions. However, not all inequalities are equivalent.
For instance, let's look at the inequalities \( x - 4 < 5 \) and \( x > 9 \).
  • The first inequality, once solved, gives a solution set of all numbers less than 9.
  • The second inequality gives a solution set of all numbers greater than 9.
Since these solution sets do not overlap and describe different ranges, the inequalities are not equivalent. Understanding whether inequalities are equivalent helps ensure that the correct comparisons and conclusions are made when working with different sets of inequalities.

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Most popular questions from this chapter

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=\frac{5 x}{x^{2}+4}\) Inequalities (a) \(y \geq 1\) (b) \(y \leq 0\)

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=\sqrt{11 x-30}-x$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$(x-3)^{2} \geq 1$$

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). Sample data from 12 athletes are shown below. (Spreadsheet at LarsonPrecalculus.com) (165,170),(184,185),(150,200), (210,255),(196,205),(240,295), (202,190),(170,175),(185,195), (190,185),(230,250),(160,150) (a) Use a graphing utility to plot the data. (b) A model for the data is \(y=1.3 x-36\).Use the graphing utility to graph the equation in the same viewing window used in part (a). (c) Use the graph to estimate the values of \(x\) that predict a maximum bench- press weight of at least 200 pounds. (d) Use the graph to write a statement about the accuracy of the model. If you think the graph indicates that an athlete's weight is not a good indicator of the athlete's maximum bench-press weight, list other factors that might influence an individual's maximum bench-press weight.

Completely factor the expression over the real numbers. $$x^{5}-27 x^{2}$$
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