Chapter 1

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I solve an equation that is quadratic in form, it's important to write down the substitution that I am making.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. 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?

In Exercises \(127-130,\) solve each equation by the method of your choice. $$ \sqrt{2} x^{2}+3 x-2 \sqrt{2}=0 $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find \(b\) such that \(\frac{7 x+4}{b}+13=x\) has a solution set given by \(\\{-6\\}\).

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. 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?

In Exercises \(127-130,\) solve each equation by the method of your choice. $$ \sqrt{3} x^{2}+6 x+7 \sqrt{3}=0 $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(\sqrt{x+4}=-5\) and \(x+4=25\) have the same solution set.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

In a round-robin chess tournament, each player is paired with every other player once. The formula $$N=\frac{x^{2}-x}{2}$$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve Exercises \(131-132\). In a round-robin chess tournament, 21 games were played. How many players were entered in the tournament?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. To solve \(x-9 \sqrt{x}+14=0,\) we let \(\sqrt{u}=x\)

Jane's salary exceeds Jim's by \(\$ 150\) per week. If \(x\) represents Jim's weekly salary, write an algebraic expression that models Jane's weekly salary.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. 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 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.

In a round-robin chess tournament, each player is paired with every other player once. The formula $$N=\frac{x^{2}-x}{2}$$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve Exercises \(131-132\). In a round-robin chess tournament, 36 games were played. How many players were entered in the tournament?

A telephone texting plan has a monthly fee of \(\$ 20\) with a charge of \(\$ 0.05\) per text. Write an algebraic that models the plan's monthly cost for \(x\) text messages.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. Parts for an automobile repair cost \(\$ 175 .\) The mechanic charges \(\$ 34\) per hour. If you receive an estimate for at least \(\$ 226\) and at most \(\$ 294\) for fixing the car, what is the time interval that the mechanic will be working on the job?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { Solve: } \sqrt{6 x-2}=\sqrt{2 x+3}-\sqrt{4 x-1}$$

If the width of a rectangle is represented by \(x\) and the length is represented by \(x+200,\) write a simplified algebraic expression that models the rectangle's perimeter.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three month pass to be the best deal?

When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { Solve for } x: \sqrt[3]{x \sqrt{x}}=9$$

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { Solve for } x: x^{\frac{5}{6}}+x^{\frac{2}{3}}-2 x^{\frac{1}{2}}=0$$

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

Will help you prepare for the material covered in the next section. Is \(-1\) a solution of \(3-2 x \leq 11 ?\)

What is a compound inequality and how is it solved?

Will help you prepare for the material covered in the next section. $$\text { Solve: }-2 x-4=x+5$$

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Will help you prepare for the material covered in the next section. $$\text { Solve: } \frac{x+3}{4}=\frac{x-2}{3}+\frac{1}{4}$$

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 6 miles long and 3 miles wide. How long is a pedestrian route that runs diagonally across the park?

Explain why \(|x|<-4\) has no solution.

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 4 miles long and 2 miles wide. How long is a pedestrian route that runs diagonally across the park?

Describe the solution set of \(|x|>-4\).

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. The base of a 30 -foot ladder is 10 feet from a building. If the ladder reaches the flat roof, how tall is the building?

Solve each inequality using a graphing utility. Graph each side separately. Then determine the values of x for which the graph for the left side lies above the graph for the right side. $$-3(x-6)>2 x-2$$

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. A baseball diamond is actually a square with 90 -foot sides. What is the distance from home plate to second base?

Solve each inequality using a graphing utility. Graph each side separately. Then determine the values of x for which the graph for the left side lies above the graph for the right side. $$-2(x+4)>6 x+16$$

An isosceles right triangle has legs that are the same length and acute angles each measuring \(45^{\circ} .\) (GRAPH NOT COPY) a. Write an expression in terms of \(a\) that represents the length of the hypotenuse. b. Use your result from part (a) to write a sentence that describes the length of the hypotenuse of an isosceles right triangle in terms of the length of a leg.

The length of a rectangular sign is 3 feet longer than the width. If the sign's area is 54 square feet, find its length and width.

A bank offers two checking account plans. Plan A has a base service charge of \(\$ 4.00\) per month plus \(10 \not c\) per check. Plan B charges a base service charge of \(\$ 2.00\) per month plus \(15 \not c\) per check. a. Write models for the total monthly costs for each plan if \(x\) checks are written. b. Use a graphing utility to graph the models in the same \([0,50,10]\) by \([0,10,1]\) viewing rectangle. c. Use the graphs (and the intersection feature) to determine for what number of checks per month plan A will be better than plan B. d. Verify the result of part (c) algebraically by solving an inequality.

A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.

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.

Each side of a square is lengthened by 3 inches. The area of this new, larger square is 64 square inches. Find the length of a side of the original square.

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.

Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 square inches. Find the length of a side of the original square.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,1\) 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.

A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width, as shown in the figure at the top of the next column. If the area of the pool and the path combined is 600 square meters, what is the width of the path?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x\) is modeled by \(|x-20| \leq 5\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement . $$(-\infty,-1] \cap[-4, \infty)=[-4,-1]$$

A machine produces open boxes using square sheets of metal. The figure illustrates that the machine cuts equal sized squares measuring 2 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 200 cubic inches, find the length and width of the open box.