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Chapter 2: Problem 17

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. $$6 x>42$$

Short Answer

Expert verified
The solution to the inequality \(6x > 42\) is \(x > 7\). On the number line, this is shown by an open dot at 7 with an arrow pointing to the right, indicating all numbers greater than 7.

Step by step solution

01

Solve the Inequality

Solving the inequality involves isolating the variable 'x'. Here the inequality is \(6x > 42\). To isolate 'x', the equation needs to be divided by 6 on both sides. This gives: \(x > 42/6\) which simplifies to \(x > 7\).
02

Draw the Number Line

A number line is a line, on which every point shows a real number. Draw a straight line. Most number lines used in schoolwork are one-dimensional, running left to right.
03

Indicate the Solution on The Number Line

On the number line, the solution \(x > 7\) means 'x' is greater than 7. So, put an open dot at 7 and draw an arrow to the right, indicating that 'x' is any number greater than 7.

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Key Concepts

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

Real Number Line
Understanding the real number line is crucial for solving inequalities. Imagine the real number line as an infinite straight path where each point corresponds to a real number. The line is typically horizontal, stretching out indefinitely in both directions.

To visualize it:
  • The left side represents smaller values, or negative numbers.
  • The right side stands for larger values, or positive numbers.
  • Every point on the line has an exact position corresponding to a real number.
The real number line is essential for graphically representing solutions to inequalities. For instance, when we solve the inequality \( 6x > 42 \), it translates to \( x > 7 \). We represent this solution on the number line from the point just greater than 7, continuing to the right, showing all numbers greater than 7.
Graphing Utility
A graphing utility is like a digital assistant for mathematics. It helps visualize mathematical concepts, including inequalities. By using software or online graphing tools, you can quickly and accurately plot equations and observe their behavior.

If you're dealing with \( 6x > 42 \), setting this up in a graphing utility can aid:
  • Confirming the solution visually by plotting \( x > 7 \).
  • Seeing the inequality displayed as a region or line on a graph.
  • Understanding relationships between numbers in the context of the inequality.
Accessing these utilities is easy and they offer features beyond manual plotting, such as zooming in for better precision, helping to reinforce your understanding of variable relationships.
Isolation of Variables
Isolation of variables is a technique used in mathematics to solve equations and inequalities. It involves rearranging the equation so that one variable stands alone, making it easier to determine its value or range of values.

Here's how you isolate a variable in the inequality \( 6x > 42 \):
  • Divide each side of the inequality by 6, which is the coefficient of \( x \).
  • This gives \( x > 7 \) after simplification.
  • Isolation helps us see that \( x \) must be greater than 7 to satisfy the inequality.
This process is straightforward but powerful because it removes extra factors, focusing solely on the variable of interest. Mastering this technique is key to solving many algebraic problems efficiently.

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Most popular questions from this chapter

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$2 x^{3}+5 x^{2}>6 x+9$$

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 \(y=-x^{2}+2 x+3\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 3\)

Meteorology A meteorologist is positioned 100 feet from the point at which a weather balloon is launched. When the balloon is at height \(h,\) the distance \(d\) (in feet) between the meteorologist and the balloon is given by \(d=\sqrt{100^{2}+h^{2}}\) (a) Use a graphing utility to graph the equation. Use the trace feature to approximate the value of \(h\) when \(d=200.\) (b) Complete the table. Use the table to approximate the value of \(h\) when \(d=200.\) $$\begin{array}{|c|c|c|c|c|c|c|} \hline h & 160 & 165 & 170 & 175 & 180 & 185 \\ \hline d & & & & & & \\ \hline \end{array}$$ (c) Find \(h\) algebraically when \(d=200.\) (d) Compare the results of each method. In each case, what information did you gain that wasn't revealed by another solution method?

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x-15|=x^{2}-15 x$$

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.
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