Question

build understanding
recall that an equation is a mathematical sentence showing that two quantities are equal, or equivalent. likewise, an inequality is a mathematical sentence showing that two quantities are not equivalent. the meanings of inequality symbols are shown in the table.

the number line shows ( x leq 365 ). note the closed circle represents \or equal to\ in the inequality to indicate the inclusion of 365.

1 a submarine is at sea level, and it descends with a rate of change of -10 feet per second.
a. write an inequality to represent the time ( t ) it takes the submarine to reach an elevation of -140 feet or deeper.

b. write three values for ( t ) that will make the inequality from part a true. substitute each value in the inequality to check. then describe the set of numbers that can make the inequality true.

c. solve the inequality from part a by writing a simple inequality that describes all of the numbers that will make the inequality true. write an inequality symbol in the first box and a number in the second box.
( t square square )
d. graph the inequality from part c. can the submarine reach an elevation of -140 feet or deeper in less than 14 seconds?

how are solving ( -10 t=-140 ) and ( -10 t leq-140 ) alike? how are they different?

Explanation:

Step1: Write the inequality for Part A

The submarine descends at a rate of - 10 feet per second. The elevation is given by \(y=-10t\) (where \(y\) is the elevation and \(t\) is the time). We want to reach an elevation of \(-140\) feet or deeper (\(y\leq - 140\)). Substituting \(y=-10t\) into the inequality, we get \(-10t\leq - 140\).

Step2: Find values for Part B

Let's pick \(t = 14\): \(-10\times14=-140\), and \(-140=-140\) (the inequality \(-10t\leq - 140\) holds).
Let \(t = 15\): \(-10\times15=-150\), and \(-150<-140\) (the inequality \(-10t\leq - 140\) holds).
Let \(t = 16\): \(-10\times16=-160\), and \(-160<-140\) (the inequality \(-10t\leq - 140\) holds). The set of numbers that make the inequality true is all non - negative real numbers \(t\) such that \(t\geq14\) (since time \(t\geq0\)).

Step3: Solve the inequality for Part C

We have \(-10t\leq - 140\). Divide both sides by \(-10\). When dividing an inequality by a negative number, the direction of the inequality sign changes. So \(\frac{-10t}{-10}\geq\frac{-140}{-10}\), which simplifies to \(t\geq14\).

Step4: Analyze the graph for Part D

The inequality \(t\geq14\) is graphed on the number line with a closed circle at \(t = 14\) (because \(t = 14\) is included in the solution set) and an arrow to the right. Since the solution of the inequality is \(t\geq14\), the submarine cannot reach an elevation of \(-140\) feet or deeper in less than \(14\) seconds.

Answer:

A. \(-10t\leq - 140\)
B. \(t = 14\), \(t = 15\), \(t = 16\); the set is \(t\geq14,t\in\mathbb{R}\) (where \(t\) is non - negative)
C. \(t\geq14\)
D. No, the submarine cannot reach an elevation of \(-140\) feet or deeper in less than \(14\) seconds.