Chapter 1

Let \((c, d)\) be any point in the plane with \(c \neq 0 .\) Prove that \((c, d)\) and \((-c,-d)\) lie on the same straight line through the origin, on opposite sides of the origin, the same distance from the origin. [Hint: Find the midpoint of the line segment joining \((c, d) \text { and }(-c,-d) .]\)

Show that the diagonals of a square are perpendicular. I Hint: Place the square in the first quadrant of the plane, with one vertex at the origin and sides on the positive axes. Label the coordinates of the vertices appropriately.]

According to data from the Center for Science in the Public Interest, the healthy weight range for a person depends on the person's height. For example, Height \(5 \mathrm{ft} 8 \mathrm{in}\) \(6 \mathrm{ft} 0 \mathrm{in}\) Healthy Weight Range ( \(\mathbf{l b}\) ) \(143 \pm 21\) \(163 \pm 26\)

Proof of the Midpoint Formula Let \(P\) and \(Q\) be the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) respectively, and let \(M\) be the point with coordinates $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$$ Use the distance formula to compute the following: (a) The distance \(d\) from \(P\) to \(Q\) (b) The distance \(d_{1}\) from \(M\) to \(P\) (c) The distance \(d_{2}\) from \(M\) to \(Q\) (d) Verify that \(d_{1}=d_{2}\) (e) Show that \(d_{1}+d_{2}=d .\) [Hint: Verify that \(d_{1}=\frac{1}{2} d\) and \(\left.d_{2}=\frac{1}{2} d .\right]\) (f) Explain why parts (d) and (e) show that \(M\) is the midpoint of \(P Q\).

At Statewide Insurance, each department's expenses are reviewed monthly. A department can fail to pass the budget variance test in a category if either (i) the absolute value of the difference between actual expenses and the budget is more than \(\$ 500\) or (ii) the absolute value of the difference between the actual expenses and the budget is more than \(5 \%\) of the budgeted amount. Which of the following items fail the budget variance test? Explain your answers. $$\begin{array}{|l|c|c|} \hline \multirow{2}{*}\text { Item } & \multirow{2}{*}\begin{array}{c} \text { Budgeted } \\ \text { Expense (\$) } \end{array} & \multirow{2}{*}\begin{array}{c} \text { Actual } \\ \text { Expense (\$) } \end{array} \\ \hline \text { Wages } & 220,750 & 221,239 \\ \hline \text { Overtime } & 10,500 & 11,018 \\ \hline \begin{array}{l} \text { Shipping and } \\ \text { Postage } \end{array} & \multirow{2}{*}530 & \multirow{2}{*}589 \\ \hline \end{array}$$

Find a number \(k\) such that the given equation has exactly one real solution. $$k x^{2}+8 x+1=0$$

The discriminant of the equation \(a x^{2}+b x+c=0\) (with \(a, b, c\) integers) is given. Use it to determine whether or not the solutions of the equation are rational numbers. $$b^{2}-4 a c=25$$

The discriminant of the equation \(a x^{2}+b x+c=0\) (with \(a, b, c\) integers) is given. Use it to determine whether or not the solutions of the equation are rational numbers. $$b^{2}-4 a c=0$$

The discriminant of the equation \(a x^{2}+b x+c=0\) (with \(a, b, c\) integers) is given. Use it to determine whether or not the solutions of the equation are rational numbers. $$b^{2}-4 a c=72$$

Write the given expression without using absolute values. $$\left|t^{2}\right|$$

Write the given expression without using absolute values. $$\left|-2-y^{2}\right|$$

Find a number \(k\) such that 4 and 1 are the solutions of \(x^{2}-5 x+k=0\).

Write the given expression without using absolute values. $$|b-3| \text { if } b \geq 3$$

Suppose \(a, b, c\) are fixed real numbers such that \(b^{2}-4 a c \geq 0 .\) Let \(r\) and \(s\) be the solutions of $$ a x^{2}+b x+c=0 $$ (a) Use the quadratic formula to show that \(r+s=-b / a\) and \(r s=c / a\) (b) Use part (a) to verify that \(a x^{2}+b x+c=\) \(a(x-r)(x-s)\) (c) Use part (b) to factor \(x^{2}-2 x-1\) and \(5 x^{2}+8 x+2\)

(a) Solve \(x^{2}+5 x+2=0\) (exact answer required). (b) If you have one of the calculators listed below, use its polynomial solver to solve the equation in part (a). Does your answer agree with the one in part (a)? Calculator (TI- \(84+\)) (TI- 86) (TI-89) (Casio 9850) (HP-39gs) Use this menu/choice (APPS/PolySmlt") (POLY) (ALGEBRA/Solve^t) (EQUATION (Main Menu) (MATH/POLYNOM/Polyroor) (c) Use the solver to solve \(3 x^{4}-2 x^{3}-5 x^{2}+2 x+1=0\)

Write the given expression without using absolute values. $$|c-d| \text { if } c \geq d$$

Write the given expression without using absolute values. $$|u-v|-|v-u|$$

Write the given expression without using absolute values. $$\frac{|u-v|}{|v-u|} \text { if } u \neq v, u \neq 0, v \neq 0$$

Explain why the given statement is true for any numbers \(c\) and \(d .\) I Hint: Look at the properties of absolute value on page 10.1 $$\left|(c-d)^{2}\right|=c^{2}-2 c d+d^{2}$$

Explain why the given statement is true for any numbers \(c\) and \(d .\) I Hint: Look at the properties of absolute value on page 10.1 $$\sqrt{9 c^{2}-18 c d+9 d^{2}}=3|c-d|$$

Express the given geometric statement about numbers on the number line algebraically, using absolute values. The distance from \(x\) to 5 is less than 4

Express the given geometric statement about numbers on the number line algebraically, using absolute values. is more than 6 units from \(c\)

Express the given geometric statement about numbers on the number line algebraically, using absolute values. \(x\) is at most 17 units from -4

Express the given geometric statement about numbers on the number line algebraically, using absolute values. \(x\) is within 3 units of 7

Express the given geometric statement about numbers on the number line algebraically, using absolute values. \(x\) is closer to 1 than to 4

Translate the given algebraic statement into a geometric statement about numbers on the number line. $$|x-3|<2$$

Translate the given algebraic statement into a geometric statement about numbers on the number line. $$|x-c|>6$$

Translate the given algebraic statement into a geometric statement about numbers on the number line. $$|x+7| \leq 3$$

Translate the given algebraic statement into a geometric statement about numbers on the number line. $$|u+v| \geq 2$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x|=1$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x-2|=1$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x+3|=2$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x+\pi|=4$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$\left|x-\frac{3}{2}\right|=5$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x|<7$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x| \geq 5$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x-5|<2$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x-6|>2$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x+2| \geq 3$$

Use the geometric approach explained in the text to solve the given equation or inequality. $$|x+4| \leq 2$$

Explain why the statement \(|a|+|b|+|c|>0\) is algebraic shorthand for "at least one of the numbers \(a, b, c,\) is different from zero."