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Chapter 3: Problem 9

Match each description in parts (a) to (e) with the appropriate compound inequality in parts (f) to (j). a. \(y\) is greater than 4 and less than \(x-3\). b. \(y\) is greater than or equal to \(x-3\) and less than -6 . c. \(y\) is less than \(2 x+5\) and greater than -6 . d. \(y\) is greater than or equal to \(2 x+5\) and less than -6 . e. \(y\) is less than or equal to \(x-3\) and greater than \(2 x+5\). f. \(2 x+5 \leq y<-6\) g. \(4

Short Answer

Expert verified
a = g, b = i, c = j, d = f, e = h.

Step by step solution

01

- Reading the descriptions

Read each description in parts (a) to (e) carefully to understand what the inequality states. For example, part (a) states that y is greater than 4 and less than x - 3.
02

- Understand the notation

Recognize that 'greater than' and 'less than' can be translated into inequalities. For example, 'y is greater than 4' translates to y > 4.
03

- Translate each description into an inequality

Translate each description into its corresponding inequality form. For example, 'greater than' translates to '>'. Use this to rewrite each description as inequalities.
04

- Match inequalities with given compound inequalities

Compare each translated inequality from Step 3 to the compound inequalities provided in parts (f) to (j). Ensure the inequalities match both parts, the lower and upper bounds.
05

- Match part (a)

For (a): 'y is greater than 4' and 'less than x - 3' translates to 4 < y < x - 3. Compare this with the provided inequalities and see that it matches 'g'. Hence, (a) matches (g).
06

- Match part (b)

For (b): 'y is greater than or equal to x - 3' and 'less than -6' translates to x - 3 鈮� y < -6. Compare this with the provided inequalities and see that it matches 'i'. Hence, (b) matches (i).
07

- Match part (c)

For (c): 'y is less than 2x + 5' and 'greater than -6' translates to -6 < y < 2x + 5. Compare this with the provided inequalities and see that it matches 'j'. Hence, (c) matches (j).
08

- Match part (d)

For (d): 'y is greater than or equal to 2x + 5' and 'less than -6' translates to 2x + 5 鈮� y < -6. Compare this with the provided inequalities and see that it matches 'f'. Hence, (d) matches (f).
09

- Match part (e)

For (e): 'y is less than or equal to x - 3' and 'greater than 2x + 5' translates to 2x + 5 < y 鈮� x - 3. Compare this with the provided inequalities and see that it matches 'h'. Hence, (e) matches (h).

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Key Concepts

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

symbolic representation
In mathematics, expressing statements using symbols is incredibly helpful for understanding and solving problems. This process is known as symbolic representation. When working with inequalities, it's important to break down sentences or descriptions into their symbolic form.
For example, consider the description: 'y is greater than 4 and less than x - 3'. Using symbols, we write this as:
  • 'y is greater than 4' becomes \( y > 4 \).
  • 'y is less than x - 3' becomes \( y < x - 3 \).
Once we gather both parts, we combine them into a single compound inequality:
\[ 4 < y < x - 3 \].
Symbols make it easier to see the relationships between variables. Practicing this helps translate between word problems and mathematical statements smoothly.

inequality notation
Inequality notation is the language used to express relationships where quantities are not equal. Understanding this notation is fundamental to working with inequalities. There are four basic inequality symbols:
  • '\( < \)' means 'less than'.
  • '\( \leq \)' means 'less than or equal to'.
  • '\( > \)' means 'greater than'.
  • '\( \geq \)' means 'greater than or equal to'.
For example, if the inequality says 'y is greater than or equal to x - 3' and 'less than -6', we can write this in inequality notation as:
\[ x - 3 \leq y < -6 \].
Notice how we connect the different parts of the inequality with conjunctions. This combined statement is known as a compound inequality. Knowing how to read and write in this notation is critical for solving and understanding problems related to inequalities.

mathematical matching
Mathematical matching in the context of inequalities means pairing the given descriptions with the corresponding compound inequalities. To do this effectively, make sure to follow these steps:
1. **Translate the description**: Turn the word-based statements into mathematical inequalities.
2. **Compare parts**: Ensure that both the lower and upper bounds are matched accurately.
For example, take the description: 'y is less than 2x + 5 and greater than -6'. First, write the inequalities:
  • 'y is less than 2x + 5' translates to \( y < 2x + 5 \).
  • 'y is greater than -6' translates to \( y > -6 \).
Combining these gives us the compound inequality:
\[ -6 < y < 2x + 5 \].
Then, find the provided option that matches this exactly. In this case, it matches with inequality (j).
Practicing this approach consistently will improve your ability to match inequalities accurately. Attention to detail is key, as the order and combination of terms must be exact.

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Most popular questions from this chapter

Create a graph for each piecewise function: a. \(f(x)=\left\\{\begin{array}{ll}2 x+1 & \text { if } x \leq 0 \\ 1 & \text { if } x>0\end{array}\right.\) b. \(g(x)=\left\\{\begin{array}{ll}-x & \text { if } x<0 \\ x & \text { if } x \geq 0\end{array}\right.\)

(Graphing program required.) A company manufactures a particular model of DVD player that sells to retailers for \(\$ 85\). It costs \(\$ 55\) to manufacture each DVD player, and the fixed manufacturing costs are \(\$ 326,000 .\) a. Create the revenue function \(R(x)\) for selling \(x\) number of DVD players. b. Create the cost function \(C(x)\) for manufacturing \(x\) DVD players. c. Plot the cost and revenue functions on the same graph. Estimate and interpret the breakeven point. d. Shade in the region where the company would make a profit. e. Shade in the region where the company would experience a loss. f. What is the inequality that represents the profit region?

The supply and demand equations for a particular bicycle model relate price per bicycle, \(p\) (in dollars) and \(q,\) the number of units (in thousands). The two equations are $$ p=250+40 q \quad \text { Supply } $$ $$ p=510-25 q $$ Demand a. Sketch both equations on the same graph. On your graph identify the supply equation and the demand equation. b. Find the equilibrium point and interpret its meaning.

(Graphing program recommended.) The blood alcohol concentration (BAC) limits for drivers vary from state to state, but for drivers under the age of 21 it is commonly set at 0.02. This level (depending upon weight and medication levels) may be exceeded after drinking only one 12 -oz can of beer. The formula $$ N=6.4+0.0625(W-100) $$ gives the number of ounces of beer, \(N,\) that will produce a BAC legal limit of 0.02 for an average person of weight \(W\). The formula works best for drivers weighing between 100 and \(200 \mathrm{lb}\). a. Write an inequality that describes the condition of too much blood alcohol for drivers under 21 to legally drive. b. Graph the corresponding equation and label the areas that represent legally safe to drive, and not legally safe to drive conditions. c. How many ounces of beer is it legally safe for a 100 -lb person to consume? A 150-lb person? A 200-lb person? d. Simplify your formula in part (b) to the standard \(y=m x+b\) form. e. Would you say that " 6 oz of beer +1 oz for every \(20 \mathrm{lb}\) over \(100 \mathrm{lb}\) " is a legally safe rule to follow?

A financial advisor has up to \(\$ 30,000\) to invest, with the stipulation that at least \(\$ 5000\) is to be placed in Treasury bonds and at most \(\$ 15,000\) in corporate bonds. a. Construct a set of inequalities that describes the relationship between buying corporate vs. Treasury bonds where the total amount invested must be less than or equal to \(\$ 30,000\). (Let \(C\) be the amount of money invested in corporate bonds, and \(T\) the amount invested in Treasury bonds.). b. Construct a feasible region of investment; that is, shade in the area on a graph that satisfies the spending constraints on both corporate and Treasury bonds. Label the horizontal axis "Amount invested in Treasury bonds" and the vertical axis "Amount invested in corporate bonds." c. Find all of the intersection points (corner points) of the bounded investment feasibility region and interpret their meanings.
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