Chapter 2

In your own words, describe the procedure for removing symbols of grouping.

What does it mean to simplify an algebraic expression?

Determine which verbal phrase(s) is (are) equivalent to the expression \(n+4\). (a) 4 more than \(n\) (b) the sum of \(n\) and 4 (c) \(n\) less than 4 (d) the ratio of \(n\) to 4 (e) the total of 4 and \(n\)

In Exercises \(75-86\), simplify the expression. $$ x^{2}-2 x y+4+x y $$

In Exercises 69 and 70, identify the variable(s) in the expression. $$ -3 \cdot(x-y) \cdot(x-y) \cdot(-3) \cdot(-3) $$

Determine whether order is important when translating each verbal phrase into an algebraic expression. Explain. (a) \(x\) increased by 10 (b) 10 decreased by \(x\) (c) The product of \(x\) and 10 (d) The quotient of \(x\) and 10

$$ r^{2}+3 r s-6-r s $$

$$ \text { In Exercises 73-76, rewrite the product in exponential form. } $$ $$ (u-v) \cdot(u-v) \cdot 8 \cdot 8 \cdot 8 \cdot(u-v) $$

Give two interpretations of "the quotient of 5 and a number times 3 ." Explain why \(\frac{3 n}{5}\) is not a possible interpretation.

$$ 5 z-5+10 z+2 z+16 $$

In Exercises 77-80, evaluate the algebraic expression for the given values of the variable(s). Area of a Triangle \(\frac{1}{2} b h\) (a) \(b=3, h=5\) (b) \(b=2, h=10\)

Give two interpretations of "the difference of 6 and a number divided by \(3 . "\) Explain why \(\frac{n-6}{3}\) is not a possible interpretation.

In Exercises \(75-86\), simplify the expression. $$ 7 x-4 x+8+3 x-6 $$

In Exercises 77-80, evaluate the algebraic expression for the given values of the variable(s). Distance Traveled \(r t\) (a) \(r=50, t=3.5\) (b) \(r=35, t=4\)

In Exercises 79-84, evaluate the expression. \((-6)(-13)\)

In Exercises \(75-86\), simplify the expression. $$ \left(7 y^{2}\right)(-3 y) $$

In Exercises 77-80, evaluate the algebraic expression for the given values of the variable(s). Volume of a Rectangular Prism lwh (a) \(l=4, w=2, h=9\) (b) \(l=100, w=0.8, h=4\)

In Exercises 79-84, evaluate the expression. $$ |4(-6)(5)| $$

In Exercises \(75-86\), simplify the expression. $$ \left(-2 t^{3}\right)\left(4 t^{2}\right) $$

In Exercises 77-80, evaluate the algebraic expression for the given values of the variable(s). Simple Interest Prt (a) \(P=1000, r=0.08, t=3\) (b) \(P=500, r=0.07, t=5\)

In Exercises 79-84, evaluate the expression. $$ \left(-\frac{4}{3}\right)\left(-\frac{9}{16}\right) $$

In Exercises \(75-86\), simplify the expression. $$ \left(\frac{2 x}{5}\right)\left(\frac{4 x}{8}\right) $$

An advertisement for a new pair of basketball shoes claims that the shoes will help you jump 6 inches higher than without shoes. (a) Let \(x\) represent the height (in inches) jumped without shoes. Write an expression that represents the height of a jump while wearing the new shoes. (b) You can jump 23 inches without shoes. How high can you jump while wearing the new shoes? (c) Your friend can jump \(20.5\) inches without shoes. How high can she jump while wearing the new shoes?

In Exercises 79-84, evaluate the expression. $$ \frac{7}{8} \div \frac{3}{16} $$

$$ \left(-\frac{6 y^{2}}{7}\right)\left(-\frac{y}{6}\right) $$

You are driving 60 miles per hour on the highway. (a) Write an expression that represents the distance you travel in \(t\) hours. (b) How far will you travel in \(2.75\) hours?

In Exercises 79-84, evaluate the expression. $$ \left|-\frac{5}{9}\right|+2 $$

In Exercises \(75-86\), simplify the expression. $$ -4(2-5 x)+3(x+6) $$

For any natural number \(n\), the sum of the numbers \(1,2,3, \ldots, n\) is equal to \(\frac{n(n+1)}{2}, \quad n \geq 1\). Verify the formula for (a) \(n=3\), (b) \(n=6\), and (c) \(n=10\).

In Exercises 79-84, evaluate the expression. $$ -7 \frac{3}{5}-3 \frac{1}{2} $$

In Exercises \(75-86\), simplify the expression. $$ 5(x+9)-2(30+4 x) $$

A convex polygon with \(n\) sides has \(\frac{n(n-3)}{2}, \quad n \geq 4\) diagonals. Verify the formula for (a) a square (two diagonals), (b) a pentagon (five diagonals), and (c) a hexagon (nine diagonals).

In Exercises 85-88, identify the property of algebra illustrated by the statement. $$ 2 a+b=b+2 a $$

In Exercises \(75-86\), simplify the expression. $$ 7-3[7-(3+x)] $$

$$ \text { Is } 3 x \text { a term of } 4-3 x \text { ? Explain. } $$

In Exercises 85-88, identify the property of algebra illustrated by the statement. $$ -4 x(1)=-4 x $$

$$ 2 x[1-(x-4)]+x(x-3) $$

Is it possible to evaluate the expression $$ \frac{x+2}{y-3} $$ when \(x=5\) and \(y=3\) ? Explain.

In Exercises 85-88, identify the property of algebra illustrated by the statement. $$ 2(c-d)=2 c-2 d $$

In Exercises 85-88, identify the property of algebra illustrated by the statement. $$ -3 y^{3}+3 y^{3}=0 $$

Error Analysis Describe and correct the error in evaluating \(y-2(x-y)\) for \(x=2\) and \(y=-4\).

In Exercises 89-96, evaluate the expression. $$ 10-(-7) $$

In Exercises 89-96, evaluate the expression. $$ 6-10-(-12)+3 $$

The remaining area of a square with side length \(x\) after a smaller square with side length \(y\) has been removed (see figure) is \((x+y)(x-y)\). (a) Show that the remaining area can also be expressed as \(x(x-y)+y(x-y)\), and give a geometric explanation for the area represented by each term in this expression. (b) Find the remaining area of a square with side length 9 after a square with side length 5 has been removed.

In Exercises 89-96, evaluate the expression. $$ -5+10-(-9)-4 $$

In Exercises 92 and 93, explain why the two expressions are not like terms. $$ \frac{1}{2} x^{2} y, \frac{5}{2} x y^{2} $$

In Exercises 89-96, evaluate the expression. $$ -(-8)+6-4-2 $$

In Exercises 89-96, evaluate the expression. $$ (-6)(-4) $$

Does the expression \([x-(3 \cdot 4)] \div 5\) change when the parentheses are removed? Does it change when the brackets are removed? Explain.

In Exercises 89-96, evaluate the expression. $$ \frac{-56}{7} $$