Chapter 1

If the quantities \(x\) and \(y\) are related by the equation \(y=3 x\),then we say that \(y\) is _______________ _______________to \(x\) and the constant of _______________ is 3

We find the "steepness," or slope, of a line passing through two points by dividing the difference in the __________ -coordinates of these points by the difference in the _________ -coordinates. So the line passing through the points \((0,1)\) and \((2,5)\) has slope __________.

Fill in the blank with an appropriate inequality sign. (a) If \(x<5,\) then \(x-3\) ________ \(2\) (b) If \(x \leq 5,\) then \(3 x\) ________ \(15\) (c) If \(x \geq 2,\) then \(-3 x\) _______ \(-6\) (d) If \(x<-2,\) then \(-x\) ________ \(2\)

The solutions of the equation \(x^{2}-2 x-3=0\) are the _____-intercepts of the graph of \(y=x^{2}-2 x-3.\)

The point that is 3 units to the right of the y-axis and 5 units below the x-axis has coordinates (_____ ,_________)

Explain in your own words what it means for an equation to model a real-world situation, and give an example.

Yes or No? If No, give a reason. (a) When you add the same number to each side of an equation, do you always get an equivalent equation? (b) When you multiply each side of an equation by the same nonzero number, do you always get an equivalent equation? (c) When you square each side of an equation, do you always get an equivalent equation?

The imaginary number \(i\) has the property that \(i^{2}=\) ______.

Which of the following are rational expressions? (a) \(\frac{3 x}{x^{2}-1}\) (b) \(\frac{\sqrt{x+1}}{2 x+3}\) (c) \(\frac{x\left(x^{2}-1\right)}{x+3}\)

(a) Using exponential notation, we can write the product \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\) as _____________ . (b) In the expression \(3^{4}\) the number 3 is called the _____________ , and the number 4 is called the _______________.

Consider the polynomial \(2 x^{5}+6 x^{4}+4 x^{3}\). (a) How many terms does this polynomial have? ____. List the terms: ____. (b) What factor is common to each term? _____. Factor the polynomial: \(2 x^{5}+6 x^{4}+4 x^{3}=\) _____.

Give an example of each of the following: (a) A natural number (b) An integer that is not a natural number (c) A rational number that is not an integer (d) An irrational number

If the quantities \(x\) and \(y\) are related by the equation \(y=\frac{3}{x}\) then we say that \(y\) is _______________ _______________ to \(x\) and the constant of _______________ is 3

A line has the equation \(y=3 x+2\). (a) This line has slope __________. (b) Any line parallel to this line has slope __________. (c) Any line perpendicular to this line has slope __________.

The solutions of the inequality \(x^{2}-2 x-3>0\) are the \(x\) -coordinates of the points on the graph of \(y=x^{2}-2 x-3\) that lie ______ the \(x\) -axis.

The distance between the points 1a, b2 and 1c, d 2 is ________ So the distance between 11, 22 and 17, 102 is _________

In the formula \(I=P r t\) for simple interest, \(P\) stands for ____ \(r\) for ____ and \(t\) for____.

What is a logical first step in solving the equation? (a) \((x+5)^{2}=64\) (b) \((x+5)^{2}+5=64\) (c) \(x^{2}+x=2\)

For the complex number \(3+4 i\) the real part is ______ and the imaginary part is ______.

To simplify a rational expression, we cancel factors that are common to the ______ and ______. So the expression $$\frac{(x+1)(x+2)}{(x+3)(x+2)}$$ simplifies to ______.

To factor the trinomial \(x^{2}+7 x+10\), we look for two integers whose product is ____ and whose sum is ____ These integers are ____ and ____ so the trinomial factors as ____.

(a) When we multiply two powers with the same base, we ________________ the exponents. So \(3^{4} \cdot 3^{5}=\) ________________ . (b) When we divide two powers with the same base, we _________________the exponents. So \(\frac{3^{5}}{3^{2}}=\) __________________ .

Complete each statement and name the property of real numbers you have used. (a) \(a b=\) ______;______Property (b) \(a+(b+c)=\) ______;______ Property (c) \(a(b+c)=\) ______;______ Property

If the quantities \(x, y,\) and \(z\) are related by the equation \(z=3 \frac{x}{y}\), then we say that \(z\) is _______________ _______________ to \(x\) and _______________ _______________ to \(y\).

The point-slope form of the equation of the line with slope 3 passing through the point \((1,2)\) is __________.

(a) The solution of the inequality \(|x| \leq 3\) is the interval ___________ (b) The solution of the inequality \(|x| \geq 3\) is a union of two intervals __________ \(U\) ___________

Explain how you would use each method to solve the equation \(x^{2}-4 x-5=0\) (a) By factoring: _______ (b) By completing the square: _______ (c) By using the Quadratic Formula: ______

Give a formula for the area of the geometric figure. (a) A square of side \(x: A=\)____ (b) A rectangle of length \(l\) and width \(w: \quad A=\)____ (c) A circle of radius \(r: A=\)____

(a) The complex conjugate of \(3+4 i\) is \(3+4 i=\) ______. (b) \((3+4 i)(\overline{3+4 i})=\) ______.

The greatest common factor in the expression \(3 x^{3}+x^{2}\) is ____ and the expression factors as ____ (____+____).

(a) Using exponential notation, we can write \(\sqrt[3]{5}\) as _____________ . (b) Using radicals, we can write \(5^{1 / 2}\) as ____________ . (c) Is there a difference between \(\sqrt{5^{2}}\) and \((\sqrt{5})^{2} ?\) Explain.

Express the set of real numbers between but not including 2 and 7 as follows. (a) In set-builder notation: ______ (b) In interval notation: ______

For the linear equation \(2 x+3 y-12=0,\) the \(x\)-intercept is __________ and the \(y\)-intercept is _________. The equation in slope-intercept form is \(y=\) ____________. The slope of the graph of this equation is ____________.

(a) The set of all points on the real line whose distance from zero is less than 3 can be described by the absolute value inequality \(|x|\) ________ . (b) The set of all points on the real line whose distance from zero is greater than 3 can be described by the absolute value inequality \(|x|\) ________ .

If the point 12, 32 is on the graph of an equation in x and y, then the equation is satisfied when we replace x by _______ and y by _______ Is the point 12, 32 on the graph of the equation \(2 y=x+1 ?\) Complete the table, and sketch a graph.

Balsamic vinegar contains \(5 \%\) acetic acid, so a 32 -oz bottle of balsamic vinegar contains ____ ounces of acetic acid.

(a) The solutions of the equation \(x^{2}(x-4)=0\) are ______ (b) To solve the equation \(x^{3}-4 x^{2}=0,\) we ________ the left-hand side.

If \(3+4 i\) is a solution of a quadratic equation with real coefficients, then ______ is also a solution of the equation.

The Special Product Formula for the "square of a sum" is \((A+B)^{2}=\) _____. So \((2 x+3)^{2}=\) _____.

Consider the expression \(\frac{1}{x}-\frac{2}{x+1}-\frac{x}{(x+1)^{2}}\). (a) How many terms does this expression have? (b) Find the least common denominator of all the terms. (c) Perform the addition and simplify.

Explain what \(4^{3 / 2}\) means, then calculate \(4^{3 / 2}\) in two different ways: \(\left(4^{1 / 2}\right)=\) ____________ or \(\left(4^{3}\right)=\) _____________

The symbol \(|x|\) stands for the ______ of the number \(x\). If \(x\) is not \(0,\) then the sign of \(|x|\) is always ______.

Each equation, is \(y\) directly proportional, inversely proportional, or not proportional to \(x ?\). (a) \(y=3 x\) (b) \(y=3 x+1\)

The slope of a horizontal line is _________. The equation the vertical line passing through \((2,3)\) is ____________.

Yes or No? If No, give an example. (a) If \(x(x+1)>0,\) does it follow that \(x\) is positive? (b) If \(x(x+1)>5,\) does it follow that \(x>5 ?\)

Solve the equation \(\sqrt{2 x}+x=0\) by doing the following steps. (a) Isolate the radical: _______ (b) Square both sides: ______ (c) The solutions of the resulting quadratic equation are ______ (d) The solution(s) that satisfy the original equation are _______

Solve the equation both algebraically and graphically. $$x-4=5 x+12$$

A painter paints a wall in \(x\) hours, so the fraction of the wall that she paints in 1 hour is ____.

Yes or No? If \(\mathrm{No},\) give a reason. Is every real number also a complex number?

The Special Product Formula for the "product of the sum and difference of terms" is \((A+B)(A-B)=\) ____. \(\operatorname{So}(5+x)(5-x)=\) _____.