Chapter 1

In Exercises 122–133, 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 dollar. The mechanic charges 34 dollar per hour. If you receive an estimate for at least 226 dollar and at most 294 dollar for fixing the car, what is the time interval that the mechanic will be working on the job?

Exercises \(131-133\) 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 Exercises 122–133, 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 dollar. A three-month pass costs 7.50 dollar and reduces the toll to 0.50 dollar. 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?

Solve: \(\sqrt{6 x-2}=\sqrt{2 x+3}-\sqrt{4 x-1}\)

Exercises \(131-133\) 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.

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

Solve without squaring both sides: \(5-\frac{2}{x}=\sqrt{5-\frac{2}{x}}\)

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

Solve for \(x: \sqrt[3]{x \sqrt{x}}=9\)

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

Solve for \(x: x^{\frac{5}{6}}+x^{\frac{2}{3}}-2 x^{\frac{1}{2}}=0\)

Explaining the Concepts. What is a compound inequality and how is it solved?

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

Explaining the Concepts. 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. Solve: \(-2 x-4=x+5\)

Use the Pythagorean Theorem and the square root property to solve. 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?

Explaining the Concepts. 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. Solve: \(\frac{x+3}{4}=\frac{x-2}{3}+\frac{1}{4}\)

Use the Pythagorean Theorem and the square root property to solve. 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?

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

Use the Pythagorean Theorem and the square root property to solve. 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?

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

Use the Pythagorean Theorem and the square root property to solve. 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?

In Exercises 142–143, 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. Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. a. A wheelchair ramp with a length of 122 inches has a horizontal distance of 120 inches. What is the ramp's vertical distance? b. Construction laws are very specific when it comes to access ramps for the disabled. Every vertical rise of 1 inch requires a horizontal run of 12 inches. Does this ramp satisfy the requirements?

In Exercises 142–143, 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 \(45^{\circ} .\) 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 dollar per month plus 10¢ per check. Plan B charges a base service charge of $2.00 per month plus 15¢ 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. 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. $$ (-\infty,-1] \cap[-4, \infty)=[-4,-1] $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ (-\infty, 3) \cup(-\infty,-2)=(-\infty,-2) $$

A machine produces open boxes using square sheets of metal. The machine cuts equal-sized squares measuring 3 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 75 cubic inches, find the length and width of the open box.

A piece of wire is 8 inches long. The wire is cut into two pieces and then each piece is bent into a square. Find the length of each piece if the sum of the areas of these squares is to be 2 square inches. (GRAPH CANNOT COPY)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$ \begin{aligned} 2 &>1 \\ 2(y-x) &>1(y-x) \\ 2 y-2 x &>y-x \\ y-2 x &>-x \\ y &>x \end{aligned} $$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

What is a quadratic equation?

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

Will help you prepare for the material covered in the first section of the next chapter. Here are two sets of ordered pairs: $$ \begin{aligned} &\text { set } 1:\\{(1,5),(2,5)\\}\\\ &\operatorname{set} 2:\\{(5,1),(5,2)\\} \end{aligned} $$ In which set is each x@coordinate paired with only one y@coordinate?

Explain how to solve \(x^{2}+6 x+8=0\) using the quadratic formula.

Graph \(y=2 x\) and \(y=2 x+4\) in the same rectangular coordinate system. Select integers for \(x,\) starting with \(-2\) and ending with 2.

How is the quadratic formula derived?

What is the discriminant and what information does it provide about a quadratic equation?

If you are given a quadratic equation, how do you determine which method to use to solve it?