Chapter 1

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.

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 _____.

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

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

True or false? (a) Adding the same number to each side of an equation always gives an equivalent equation. (b) Multiplying each side of an equation by the same number always gives an equivalent equation. (c) Squaring each side of an equation always gives an equivalent equation.

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

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

Consider the polynomial \(2 x^{5}+6 x^{4}+4 x^{3}\) How many terms does this polynomial have?___ List the terms:____ 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

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

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.

True or false? (a) If \(x(x+1)>0,\) then \(x\) and \(x+1\) are either both positive or both negative. (b) If \(x(x+1)>5,\) then \(x\) and \(x+1\) are each greater than 5

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

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.

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 _____.

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:_______

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

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 \(y.\)

(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 ___________ \(\bigcup\) _________.

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

To multiply two rational expressions, we multiply their _____ together and multiply their _____ together. So $$\frac{2}{x+1} \cdot \frac{x}{x+3}$$ is the same as _____.

(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.

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 \(w: A=\) _______. (c) A circle of radius \(r: A=\) _______.

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

The set of numbers between but not including 2 and 7 can be written as follows: ______ in set-builder notation and _____ in interval notation.

(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.

(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 \(|x|\) __________.

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

If the point (2, 3) 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 \(2 y=x+1 ?\)

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.

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 _______

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

The Special Product Formula for the "sum and difference of the same terms" is \((A+B)(A-B)=\)______ $$\operatorname{So}(5+x)(5-x)=$$

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

Write an equation that expresses the statement. \(T\) varies directly as \(x\)

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$3-2 x \leq \frac{1}{2}$$

Find the slope of the line through \(P\) and \(Q .\) $$P(0,0), Q(4,2)$$

Find the domain of the expression. $$4 x^{2}-10 x+3$$

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

The Special Factoring Formula for the "difference of squares" is \(A^{2}-B^{2}=\)______ \(-\) So \(4 x^{2}-25\) factors as_______.

List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers $$\left\\{0,-10,50, \frac{22}{7}, 0.538, \sqrt{7}, 1.2 \overline{3},-\frac{1}{3}, \sqrt{2}\right\\}$$

Write an equation that expresses the statement. \(P\) is directly proportional to \(w\)

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$2 x-1 \geq x$$

Find the slope of the line through \(P\) and \(Q .\) $$P(0,0), Q(2,-6)$$

The graph of the equation \((x-1)^{2}+(y-2)^{2}=9\) is a circle with center (_____),(______) and radius ______