Chapter 1

Solve each inequality analytically. Write the solution set in interval notation. Support the answer graphically. $$8(4-3 x) \geq 6(6-4 x)$$

Suppose that \(P\) is an endpoint of a segment \(P Q\) and \(M\) is the midpoint of \(P Q .\) Find the coordinates of endpoint \(Q\). $$P(-10.32,8.55), M(1.55,-2.75)$$

The table lists poverty-level income cutoffs for a family of four for selected years. Use the midpoint formula to estimate the poverty-level cutoffs (rounded to the nearest dollar) in 2012 and 2014. $$\begin{aligned} &\\\ &\begin{array}{c|c} \text { Year } & \text { Income (in dollars) } \\ 2011 & 22,350 \\ 2013 & 23.550 \\ 2015 & 24,250 \end{array} \end{aligned}$$

Estimated and projected enrollments in two-year colleges for 2016 \(2018,\) and 2020 are shown in the table. Use the midpoint formula to estimate the enrollments to the nearest thousand for 2017 and 2019. $$\begin{array}{c|c} \text { Year } & \text { Enrollment (in thousands) } \\ 2016 & 7194 \\ 2018 & 7500 \\ 2020 & 7706 \end{array}$$

Triangles can be classified by their sides. (a) An isosceles triangle has at least two sides of equal length. Determine whether the triangle with vertices \((0,0),(3,4),\) and \((7,1)\) is isosceles. (b) An equilateral triangle has all sides of equal length. Determine whether the triangle with vertices \((-1,-1),(2,3),\) and \((-4,3)\) is equilateral. (c) Determine whether a triangle having vertices \((-1,0),(1,0)\) and \((0, \sqrt{3})\) is isosceles, equilateral, or neither. (d) Determine whether a triangle having vertices \((-3,3),(-2,5)\) and \((-1,3)\) is isosceles, equilateral, or neither.

Solve each compound inequality analytically. Support your answer graphically. $$4>6 x+5>-1$$

Prove that the midpoint \(M\) of the line segment joining endpoints \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) has coordinates $$ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) $$ by showing that the distance between \(P\) and \(M\) is equal to the distance between \(M\) and \(Q\) and that the sum of these distances is equal to the distance between \(P\) and \(Q\).

Solve each compound inequality analytically. Support your answer graphically. $$-3 \leq \frac{x-4}{-5}<4$$

Solve each compound inequality analytically. Support your answer graphically. $$1<\frac{4 x-5}{-2}<9$$

Solve each compound inequality analytically. Support your answer graphically. $$-\frac{1}{2}

Solve each compound inequality analytically. Support your answer graphically. $$-\frac{3}{4}<2 x-1<\frac{3}{4}$$

Solve each compound inequality analytically. Support your answer graphically. $$-4 \leq \frac{1}{2} x-5 \leq 4$$

Solve each compound inequality analytically. Support your answer graphically. $$-2<\frac{x-4}{6}<2$$

Solve each compound inequality analytically. Support your answer graphically. $$\sqrt{2} \leq \frac{2 x+1}{3} \leq \sqrt{5}$$

Solve each compound inequality analytically. Support your answer graphically. $$\pi \leq 5-4 x<7 \pi$$

Vehicle Sales In 2009 new motor vehicle sales in the United States were \(10,602\) thousand. In 2013 the figure had increased to \(15,844\) thousand. (a) Find a linear function \(P(x)=a x+b\) that models the number of vehicle sales \(x\) years after 2009 . (b) Interpret the slope of the graph of \(y=P(x)\) (c) Use \(P(x)\) to approximate the number of vehicle sales in 2011 (d) Assuming the model continued past 2013 , what would be the number of sales in \(2015 ?\)

Suppose that an aluminum can is manufactured so that its radius \(r\) can vary from 0.99 inches to 1.01 inches. What range of values is possible for the circumference \(C\) of the can? Express the answer by using a compound inequality. (IMAGE CAN NOT COPY)

Suppose that a square picture frame has sides that vary between 9.9 inches and 10.1 inches. What range of values is possible for the perimeter \(P\) of the picture frame? Express the answer by using a compound inequality.

The solution set of a linear equation is closely related to the solution set of a linear inequality.In order to investigate this connection. Write answers in interval notation when appropriate. Use the \(x\) -intercept method to find the solution set of \(3.7 x-11.1=0 .\) How many solutions are there? How many solutions are there to any conditional linear equation in one variable?

The solution set of a linear equation is closely related to the solution set of a linear inequality.In order to investigate this connection. Write answers in interval notation when appropriate. Use the \(x\) -intercept method to find the solution set of the equation $$-4 x+6=0$$ Then find the solution sets of the inequalities $$-4 x+6<0 \text { and }-4 x+6>0$$

The solution set of a linear equation is closely related to the solution set of a linear inequality.In order to investigate this connection. Write answers in interval notation when appropriate. Generalize your results from Exercises \(119-121\) by answering the questions that follow. (a) What is the solution set of \(a x+b=0\) if \(a \neq 0 ?\) (b) Suppose \(a>0 .\) What are the solution sets of \(a x+b<0\) and \(a x+b>0 ?\) (c) Suppose \(a<0 .\) What are the solution sets of \(a x+b<0\) and \(a x+b>0 ?\)