Chapter 2

Exercises \(1-8:\) Let \(a \neq 0\) Solve \(|x|=3\)

Express the following in interval notation. $$ x<2 $$

Find the point-slope form of the line passing through the given points. Use the first point as \(\left(x_{1}, y_{1}\right) .\) Plot the points and graph the line by hand. $$ (1,2),(3,-2) $$

How many solutions are there to \(a x+b=0\) with \(a \neq 0 ?\)

Functions as Models U.S. Vzhicle Production In 2000 there were 12.8 million vehicles produced in the United States, and in 2004 there were 12.0 million. The formula \(V(t)=-0.2 t+12.8\) models these data exactly, where \(t=0\) corresponds to \(2000, t=1\) to \(2001,\) and so on. (a) Verify that \(V(t)\) gives the exact values in millions for 2000 and 2004 (b) Use \(V(t)\) to estimate the number of vehicles manufactured in 2002 and 2006 . Do these estimates involve interpolation or extrapolation? (c) The actual value for 2002 was 12.3 million and for 2006 was 11.3 million. Discuss the accuracy of your results from part (b).

Exercises \(1-8:\) Let \(a \neq 0\) $$ \text { Solve }|x| \leq 3 $$

Express the following in interval notation. $$ x>-3 $$

Find the point-slope form of the line passing through the given points. Use the first point as \(\left(x_{1}, y_{1}\right) .\) Plot the points and graph the line by hand. $$ (-2,3),(1,0) $$

How many times does the graph of \(y=a x+b\) with \(a \neq 0\) intersect the \(x\) -axis?

Functions as Models U.S. Advertising Expenditures In 2002 S237 billion was spent on advertising in the United States, and in 2004 this amount was \(\$ 264\) billion. The formula \(A(t)=13.5 t+237\) models these data exactly, where \(t=0\) corresponds to \(2002, t=1\) to \(2003,\) and so on. (a) Verify that \(A(t)\) gives the exact values in billions of dollars for 2002 and 2004 (b) Use \(A(t)\) to estimate the advertising expenditures in 2000 and \(2003 .\) Do these estimates involve interpolation or extrapolation? (c) The actual value for 2000 was \(\$ 244\) billion and for 2003 was \(\$ 245\) billion. Discuss the accuracy of your results from part (b).

Exercises \(1-8:\) Let \(a \neq 0\) Solve \(|x|>3\)

Express the following in interval notation. $$ x \geq-1 $$

Apply the distributive property to \(4-(5-4 x)\)

Find the point-slope form of the line passing through the given points. Use the first point as \(\left(x_{1}, y_{1}\right) .\) Plot the points and graph the line by hand. $$ (-3,-1),(1,2) $$

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=5 x-2 $$ $$ \begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ y & 3 & 8 & 13 & 18 \end{array} $$

Express the following in interval notation. $$ x \leq 7 $$

What property is used to solve 15x = 5?

Find the point-slope form of the line passing through the given points. Use the first point as \(\left(x_{1}, y_{1}\right) .\) Plot the points and graph the line by hand. $$ (-1,2),(-2,-3) $$

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=1-0.2 x $$ $$ \begin{array}{ccccc} x & 5 & 10 & 15 & 20 \\ y & 0 & -1 & -2 & -4 \end{array} $$

Exercises \(1-8:\) Let \(a \neq 0\) Describe the graph of \(y=|a x+b|\)

Express the following in interval notation. $$ \\{x | 1 \leq x<8\\} $$

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Slope \(-2.4,\) passing through \((4,5)\)

If \(f(x)=a x+b\) with \(a \neq 0,\) how are the zero of \(f\) and the \(x\) -intercept of the graph of \(f\) related?

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=3.7-1.5 x $$ $$ \begin{array}{cccc} x & -6 & 0 & 1 \\ y & 12.7 & 3.7 & 2.1 \end{array} $$

Exercises \(1-8:\) Let \(a \neq 0\) $$ \text { Solve }|a x+b|=0 $$

Express the following in interval notation. $$ \\{x |-2

Distinguish between a contradiction and an identity.

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Slope 1.7 , passing through \((-8,10)\)

Exercises \(3-6:\) A function \(f\) is given. Determine whether \(f\) models the data exactly or approximately. $$ f(x)=13.3 x-6.1 $$ $$ \begin{array}{cccc} x & 1 & 2 & 5 \\ y & 7.2 & 20.5 & 60.4 \end{array} $$

Exercises \(1-8:\) Let \(a \neq 0\) Rewrite \(\sqrt{36 a^{2}}\) by using an absolute value.

Exercises \(7-12:\) Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 3 x-1.5=7 $$

Express the following in interval notation. $$ \\{x | x \leq 1\\} $$

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Passing through \((1,-2)\) and \((-9,3)\)

Exercises \(7-10:\) Find the formula for a linear function \(f\) that models the data in the table exactly. $$ \begin{array}{rrrr} x & -2 & 0 & 4 \\ f(x) & 4 & 3 & 1 \end{array} $$

Exercises \(1-8:\) Let \(a \neq 0\) Rewrite \(\sqrt{(a x+b)^{2}}\) by using an absolute value.

Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 100-23 x=20 x $$

Express the following in interval notation. $$ \\{x | x>5\\} $$

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Passing through \((-6,10)\) and \((5,-12)\)

Exercises \(7-10:\) Find the formula for a linear function \(f\) that models the data in the table exactly. $$ \begin{array}{rlrl} x & -6 & 0 & 3 \\ f(x) & -5 & -1 & 1 \end{array} $$

Graph by hand. (a) Find the \(x\) -intercept. (b) Determine where the graph is increasing and where it is decreasing. $$y=|x+1|$$

Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 2 \sqrt{x}+2=1 $$

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 2 x+6 \geq 10 $$

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. \(\mathbf{x}\) -intercept \(4, y\) -intercept \(-3\)

Exercises \(7-10:\) Find the formula for a linear function \(f\) that models the data in the table exactly. $$ \begin{array}{rrrr} x & 1 & 2 & 3 \\ f(x) & 7 & 9 & 11 \end{array} $$

Graph by hand. (a) Find the \(x\) -intercept. (b) Determine where the graph is increasing and where it is decreasing. $$ y=|1-x| $$

Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 4 x^{3}-7=0 $$

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ -4 x-3<5 $$

Exercises \(7-10:\) Find the formula for a linear function \(f\) that models the data in the table exactly. $$ \begin{array}{rrrr} x & 15 & 30 & 45 \\ f(x) & 40 & 30 & 20 \end{array} $$

Graph by hand. (a) Find the \(x\) -intercept. (b) Determine where the graph is increasing and where it is decreasing. $$ y=|2 x-3| $$

Determine whether the equation is linear or nonlinear by trying to write it in the form ax \(+b=0\) $$ 7 x-5=3(x-8) $$