Question

The world’s heaviest flying bird is the great bustard. A male bustard can be up to 4 feet long and weigh up to 40 pounds.

1. Write inequalities to describe the ranges of lengths and weights of male bustards.
2. Male bustards are usually about four times as heavy females. Write and solve an inequality that describes the range of weights of female bustards.

Short answer

a.The required inequality for lengths is$0<l\le 4$and the required inequality for weights is $0<w\le 40$.

b.The required inequality for weight of female bustard is $0<w_f\le 10$.

Step-by-step solution

Part a. Step 1. State the concept for inequality.

To graph the endpoint with strict inequality < or >, use the open parenthesis or a hollow circle at that endpoint.

To graph the endpoint with inequality symbol $\le \text{or}\ge$, use the bracket or a solid circle at that endpoint.

Properties of inequality:

$$\begin{gathered} 1.b\le c\Rightarrow b\pm a\le c\pm a \\ \begin{array}{l}2.b\le c\Rightarrow ab\le ac,\text{if}a>0\\ 3.b\le c\Rightarrow ab\ge ac,\text{if}a<0\end{array} \end{gathered}$$

Part a. Step 2. State the inequality.

Since, the length of the male bustard can be upto 4 feet long. This means that the male bustard can be 4 feet long and therefore use less than and equal to sign. Take las the length of a male bustard, write the inequality as,

$l\le 4$

Also, the length of the bustard’s cannot be 0 or less than 0, so length of bustard is strictly greater than 0. Therefore, the inequality is,

$0<l\le 4$

Also, the weight of a male bustard can be upto 40 pounds. This means that a male bustard can be of 40 pound weight and therefore use less than or equal to sign. Take was the weight of a male bustard, write the inequality as,

$w_m\le 40$

Also, the length of the bustard’s cannot be 0 or less than 0, so length of bustard is strictly greater than 0. Therefore, the inequality is,

$0<w_m\le 40$

Part a. Step 3. Conclusion.

Thus, the inequality for l less than or equal to 4 but strictly greater than 0, thus the solution of given inequality in interval form is, as $\left(0,4\right]$. And the inequality for$w_m$ less than or equal to 40 but strictly greater than 0, thus the solution of given inequality in interval form is, as $\left(0,40\right]$.

Part b. Step 1. State the concept for inequality.

To graph the endpoint with strict inequality < or >, use the open parenthesis or a hollow circle at that endpoint.

To graph the endpoint with inequality symbol $\le \text{or}\ge$, use the bracket or a solid circle at that endpoint.

Properties of inequality:

$$\begin{gathered} 1.b\le c\Rightarrow b\pm a\le c\pm a \\ \begin{array}{l}2.b\le c\Rightarrow ab\le ac,\text{if}a>0\\ 3.b\le c\Rightarrow ab\ge ac,\text{if}a<0\end{array} \end{gathered}$$

Part b. Step 2. Solve the relationship.

Since, the male bustard can be 4 times as heavy as females in weights. This means that the weight of females bustard are$\frac{1}{4}$ times the weight of males. Therefore, the relation between the weight of female and male bustards is,

$$\begin{gathered} w_m=4w_f \\ {w}_f=\frac{1}{4}{w}_m \end{gathered}$$

Part b. Step 3. Write the inequality.

Put the value of$w_m$ in the inequality $0<w_m\le 40$.

$$\begin{gathered} 0<w_m\le 40 \\ \frac{0}{4}<\frac{w_m}{4}\le \frac{40}{4} \\ 0<w_f\le 10 \end{gathered}$$

Part b. Step 4. Conclusion.

Thus, the inequality for$w_f$ less than or equal to 10 but strictly greater than 0, thus the solution of given inequality in interval form is, as $\left(0,10\right]$.