Chapter 1

When the sum of 6 and twice a positive number is subtracted from the square of the number, 0 results. Find the number.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After squaring both sides of a radical equation, the only solution that I obtained was extrancous, so \(\varnothing\) must be the solution set of the original equation.

When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The equation \(5 x^{\frac{2}{3}}+11 x^{\frac{1}{3}}+2-0\) is quadratic in form, but when I reverse the variable terms and obtain \(11 x^{3}+5 x^{3}+2-0,\) the resulting equation is no longer quadratic in form.

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(a\) and \(b\) are any real numbers, then \(a x+b=0\) always has one number in its solution set.

Solve each equation by the method of your choice. $$\frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}$$

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?

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.

Solve each equation by the method of your choice. $$\frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6}$$

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?

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 \(|-6|\)

Solve each equation by the method of your choice. $$\sqrt{2} x^{2}+3 x-2 \sqrt{2}=0$$

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?

Solve each equation by the method of your choice. $$\sqrt{3} x^{2}+6 x+7 \sqrt{3}=0$$

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 \(\mathrm{B}\) in the course. Describe the grades on the final that will cause this to happen.

will help you prepare for the material covered in the next section. 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.

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?

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?

will help you prepare for the material covered in the next section. A telephone texting plan has a monthly fee of \(\$ 20\) with a charge of \(\$ 0.05\) per text. Write an algebraic expression that models the plan's monthly cost for \(x\) text messages.

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?

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 threemonth pass to be the best deal?

will help you prepare for the material covered in the next section. 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.

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

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 than 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. Solve without squaring both sides: $$\text { Solve for } x: x^{6}+x^{3}-2 x^{2}-0$$

What is a compound inequality and how is it solved?

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

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

This 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.

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

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

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplificd 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?

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

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplificd 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?

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

Use the Pythagorean Theorem and the square root property to solve Exercises \(140-143 .\) Express answers in simplificd 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?

A bank offers two checking account plans. Plan A has a base service charge of \(\$ 4.00\) per month plus 10 ç per check. Plan \(\mathrm{B}\) charges a base service charge of \(\$ 2.00\) per month plus \(15 \phi\) 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.

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.

Determine whether 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.

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 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 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 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.

Each side of a square is lengthencd 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 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 \(|x-20| \leq 5\).