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Chapter 17: Problem 7

Solve the given inequalities. Graph each solution. $$\frac{1}{2} x<32$$

Short Answer

Expert verified
The solution is \( x < 64 \). Graphically, it's an open circle on 64 with a shaded line extending to the left.

Step by step solution

01

Write the Inequality

Start by writing the given inequality. We have \( \frac{1}{2} x < 32 \).
02

Isolate the Variable

To solve for \( x \), we need to get rid of the fraction. Multiply both sides of the inequality by 2 to isolate \( x \). This gives us: \[2 \times \frac{1}{2} x < 2 \times 32\]Simplifying both sides, we get:\[x < 64\]
03

Graph the Solution

To graph the solution, draw a number line. Identify the number 64 and draw an open circle at this point since 64 is not included in the solution set. Shade the number line to the left of 64 to indicate all numbers less than 64 are solutions.

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Key Concepts

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

Graphing Inequalities
When graphing inequalities, the goal is to visually represent the solutions on a number line or coordinate plane. In our example, once we have determined that the solution is all numbers less than 64, we'll use the number line.
Draw a number line and find the point that represents the number 64. Use an open circle at this point. An open circle means that 64 itself is not included in the set of solutions. This visual cue is vital as it distinguishes whether the endpoint is part of the solution or not.
Next, to complete the graph, shade the line to the left of this open circle. This shading signifies that all numbers less than 64 are part of the solution.
  • An open circle is used for inequalities such as "<" or ">".
  • A closed circle is for inequalities including equality such as "鈮�" or "鈮�".
  • Shading direction: left for less than, right for greater than.
Isolation of Variables
Isolation of variables is a key step in solving equations and inequalities. The idea is to get the variable, in this case, \( x \), by itself on one side of the inequality.
In the given exercise, \( \frac{1}{2}x < 32 \), we are dealing with a fraction involving \( x \). To isolate \( x \), we need to eliminate the fraction. Multiplying both sides by 2 will do just that.
Performing this operation:
  • Multiply both sides of the inequality by the denominator (which is 2).
  • This step scales all parts equally, maintaining the balance of the inequality.
  • Once simplified, this will provide \( x < 64 \).
By isolating \( x \), we convert the original statement into a simpler, more useful form. This process is crucial for determining the values \( x \) can take.
Inequality Solutions
Solving inequalities involves more than just throwing numbers around. It requires understanding the difference between types of inequalities and how to treat them properly.
In the exercise \( \frac{1}{2}x < 32 \), we determined that \( x < 64 \). This tells us the set of numbers that satisfy this inequality. Inequalities are not like equations; they typically have an infinite number of solutions.
Key points to remember:
  • Inequalities express a range, not just a single solution.
  • Solutions can include all numbers within a given range, represented by open or closed circles depending on whether the numbers at the endpoints are included.
  • Always remember to flip the inequality sign when multiplying or dividing both sides by a negative number (though not applied here).
Inequality solutions are fundamental in understanding how different values can satisfy conditions within mathematical and real-world contexts.

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Most popular questions from this chapter

solve the given problems. Explain the error in the following "proof" that \(3<2\) (1) \(1 / 8<1 / 4 \quad\) (2) \(0.5^{3}<0.5^{2}\)(3) \(\log 0.5^{3}<\log 0.5^{2}\)(5) \(3<2\)(4) \(3 \log 0.5<2 \log 0.5\)

Solve the given problems by setting up and solving appropriate inequalities. Graph each solution. A rectangular PV (photovoltaic) solar panel is designed to be \(1.42 \mathrm{m}\) long and supply \(130 \mathrm{W} / \mathrm{m}^{2}\) of power. What must the width of the panel be in order to supply between \(100 \mathrm{W}\) and \(150 \mathrm{W} ?\)

answer the given questions about the inequality \(0 < a < b\) \text { Is }|a-b| < b-a ?

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated system of inequalities. A rectangular computer chip is being designed such that its perimeter is no more than \(15 \mathrm{mm}\), its width at least \(2 \mathrm{mm}\) and its length at least \(3 \mathrm{mm}\). Graph the possible values of the width \(w\) and the length \(l\)

Use inequalities involving absolute values to solve the given problems. The voltage \(V\) in a certain circuit is given by \(V=6.0-200 i\) where \(i\) is the current (in \(\mathrm{A}\) ). For what values of the current is the absolute value of the voltage less than \(2.0 \mathrm{V} ?\)
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