Chapter 2

The formula $$p=15+\frac{5 d}{11}$$ describes the pressure of sea water, \(p,\) in pounds per square foot, at a depth of d feet below the surface. Use the formula to solve. The record depth for breath-held diving, by Francisco Ferreras (Cuba) off Grand Bahama Island, on November \(14,1993,\) involved pressure of 201 pounds per square foot. To what depth did Ferreras descend on this illadvised venture? (He was underwater for 2 minutes and 9 seconds!

Use properties of inequality to rewrite each inequality so that \(x\) is isolated on one side. $$-2 x-a \leq b$$

Exercises \(91-93\) will help you prepare for the material covered in the next section. Is 6 a solution of \(2(x-3)-17=13-3(x+2) ?\)

The formula $$p=15+\frac{5 d}{11}$$ describes the pressure of sea water, \(p,\) in pounds per square foot, at a depth of d feet below the surface. Use the formula to solve. At what depth is the pressure 20 pounds per square foot?

Use properties of inequality to rewrite each inequality so that \(x\) is isolated on one side. $$y \leq m x+b \text { and } m<0$$

Solve for \(x: \frac{x}{2}+7=13-\frac{x}{4}\) (Section \(2.3,\) Example 4 )

Exercises \(91-93\) will help you prepare for the material covered in the next section. Multiply and simplify: \(10\left(\frac{x}{5}-\frac{39}{5}\right)\)

In your own words, describe how to solve a linear equation.

Use properties of inequality to rewrite each inequality so that \(x\) is isolated on one side. $$y>m x+b \text { and } m>0$$

Simplify: \(\left[3\left(12 \div 2^{2}-3\right)^{2}\right]^{2}\) (Section \(1.8,\) Example 8 )

Explain how to solve a linear equation containing fractions.

We know that \(|x|\) represents the distance from 0 to \(x\) on a number line. Use each sentence to describe all possible locations of \(x\) on a number line. Then rewrite the given sentence as an inequality involving \(|x|\). The distance from 0 to \(x\) on a number line is less than 2 .

Will help you prepare for the material covered in the next section. Is 2 a solution of \(x+3<8 ?\)

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}=1\) by multiplying both sides by \(20,\) rather than the least common denominator, \(10 .\) Describe what happens. If you get the correct solution, why do you think we clear the equation of fractions by multiplying by the least common denominator?

We know that \(|x|\) represents the distance from 0 to \(x\) on a number line. Use each sentence to describe all possible locations of \(x\) on a number line. Then rewrite the given sentence as an inequality involving \(|x|\). The distance from 0 to \(x\) on a number line is less than 3 .

Will help you prepare for the material covered in the next section. Is 6 a solution of \(4 y-7 \geq 5 ?\)

Explain how to clear decimals in a linear equation.

We know that \(|x|\) represents the distance from 0 to \(x\) on a number line. Use each sentence to describe all possible locations of \(x\) on a number line. Then rewrite the given sentence as an inequality involving \(|x|\). The distance from 0 to \(x\) on a number line is greater than 2 .

Will help you prepare for the material covered in the next section. Solve: \(2(x-3)+5 x=8(x-1)\)

Suppose you are an algebra teacher grading the following solution on an examination: $$\begin{aligned}\text { Solve: } &-3(x-6) &=2-x \\\\\text { Solution: } &-3 x-18 &=2-x \\\&-2 x-18 &=2 \\\\-2 x &=&-16 \\\x &=8\end{aligned}$$ You should note that 8 checks, and the solution is \(8 .\) The student who worked the problem therefore wants full credit. Can you find any errors in the solution? If full credit is 10 points, how many points should you give the student? Justify your position.

We know that \(|x|\) represents the distance from 0 to \(x\) on a number line. Use each sentence to describe all possible locations of \(x\) on a number line. Then rewrite the given sentence as an inequality involving \(|x|\). The distance from 0 to \(x\) on a number line is greater than 3 .

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. Although I can solve \(3 x+\frac{1}{5}=\frac{1}{4}\) by first subtracting \(\frac{1}{5}\) from both sides, I find it easier to begin by multiplying both sides by \(20,\) the least common denominator.

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. I can use any common denominator to clear an equation of fractions, but using the least common denominator makes the arithmetic easier.

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. When I substituted 5 for \(x\) in the equation $$4 x+6=6(x+1)-2 x$$ I obtained a true statement, so the equation's solution is 5 .

I cleared the equation \(0.5 x+8.3=12.4\) of decimals by multiplying both sides by \(100 .\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(3(x+4)=3(4+x)\) has precisely one solution.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(x+\frac{1}{3}=\frac{1}{2}\) is equivalent to \(x+2=3\).

A woman'sheight, \(h\), is related to the length of her femur, \(f\) (the bone from the knee to the hip socket), by the formula \(f=0.432 h-10.44 .\) Both \(h\) and \(f\) are measured in inches. A partial skeleton is found of a woman in which the femur is 16 inches long. Police find the skeleton in an area where a woman slightly over 5 feet tall has been missing for over a year. Can the partial skeleton be that of the missing woman? Explain.

On two examinations, you have grades of 86 and 88 . There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of \(\mathrm{A}\), meaning a final average of at least 90 . a. What must you get on the final to earn an A in the course? b. By taking the final, if you do poorly, you might risk the \(B\) that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(B\) in the course. Describe the grades on the final that will cause this to happen.

Solve equation. \(\frac{2 x-3}{9}+\frac{x-3}{2}=\frac{x+5}{6}-1\)

On three examinations, you have grades of \(88,78,\) and 86 There is still a final examination, which counts as one grade. a. In order to get an A, your average must be at least 90 . If you get 100 on the final, compute your average and determine if an A in the course is possible. b. To earn a B in the course, you must have a final average of at least \(80 .\) What must you get on the final to earn a \(\mathrm{B}\) in the course?

Solve equation. \(2(3 x+4)=3 x+2[3(x-1)+2]\)

A car can be rented from Continental Rental for \(\$ 80\) per week plus 25 cents for each mile driven. How many miles can you travel if you can spend at most \(\$ 400\) for the week?

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. \(-24 \quad-20 \text { (Section } 1.3, \text { Example } 6)\)

A car can be rented from Basic Rental for \(\$ 60\) per week plus 50 cents for each mile driven. How many miles can you travel if you can spend at most \(\$ 600\) for the week?

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. \(-\frac{1}{3} \quad-\frac{1}{5}(\text { Section } 1.3, \text { Example } 6)\)

An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Simplify: \(-9-11+7-(-3) .\) (Section 1.6, Example 3)

An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Consider the formula $$T=D+p m$$ a. Subtract \(D\) from both sides and write the resulting formula. b. Divide both sides of your formula from part (a) by \(p\) and write the resulting formula.

When graphing the solutions of an inequality, what is the difference between a parenthesis and a bracket?

Solve: \(4=0.25 B\).

When solving an inequality, when is it necessary to change the direction of the inequality symbol? Give an example.

Solve: \(\quad 1.3=P \cdot 26\)

Describe ways in which solving a linear inequality is similar to solving a linear equation.

Describe ways in which solving a linear inequality is different from solving a linear equation.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I prefer interval notation over set-builder notation because it takes less space to write solution sets.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can check inequalities by substituting 0 for the variable: When 0 belongs to the solution set, I should obtain a true statement, and when 0 does not belong to the solution set, I should obtain a false statement.

In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

I solved \(-2 x+5 \geq 13\) and concluded that \(-4\) is the greatest integer in the solution set.