Chapter 4

Fill in the blanks. $$\left\\{\begin{array}{l}x+y \leq 2 \\\x-3 y>10\end{array}\right.$$ is a system of linear _____ in two variables.

Fill in the blanks. \(4 x-2 y \geq-8\) is an example of a _____ inequality in ____ variables.

The _______ ______ of a number is its distance from 0 on a number line.

Fill in the blanks. The ______ of two sets is the set of elements that are common to both sets and the ______ of two sets is the set of elements that are in one set, or the other, or both.

Fill in the blanks. \(<,>, \leq,\) and \(\geq\) are _____ symbols.

Fill in the blanks. To solve a system of inequalities by graphing, we graph each inequality. The solution is the region where the graphs overlap or ______.

Fill in the blanks. \(x \geq 3\) and \(x<4\) is a ______ inequality.

Fill in the blanks. \(3 x+2 \geq 7\) is an example of a _____ inequality in one variable.

The graph of a linear inequality in two variables is a region of the coordinate plane on one side of a _____ line.

Fill in the blanks. \(-6

Fill in the blanks. The graph of a set of real numbers that is a portion of a number line is called an _____.

When we graph a system of two linear inequalities, any point in the doubly shaded region has coordinates that _____ both inequalities.

Fill in the blanks. \((2,8)\) is an example of an open ______ \(,[-4,0]\) is an example of a ______ interval, and (0, 9] is an example of a half-______ interval.

Fill in the blanks. In \((-\infty, 5),\) the right _____ is used to show that 5 is not included in the interval. \(\operatorname{In}[12, \infty),\) the left _____ is used to show that 12 is included in the interval.

Check to determine whether each point satisfies the following system of linear inequalities: $$\left\\{\begin{array}{l}x+y \leq 2 \\\x-3 y>10\end{array}\right.$$ a. \((2,-3)\) b. \((12,-1)\) c. \((0,-3)\) d. \((-0.5,-5)\)

Fill in the blanks. a. The solution set of a compound inequality containing the word and includes all numbers that make ______ inequalities true. b. The solution set of a compound inequality containing the word or includes all numbers that make ______ , or the other, or ______ inequalities true.

When two equations are joined by the word or, such as \(x+1=5\) or \(x+1=-5,\) we call the statement a _________ equation.

Check to determine whether each ordered pair is a solution of \(3 x-2 y \geq 5\) a. \((3,1)\) b. \((0,3)\) c. \((-1,-4)\) d. \(\left(1, \frac{1}{2}\right)\)

Fill in the blanks. We read the set- _____ notation \(\\{x | x<1\\}\) as "the set of all real numbers \(x\) _____ _____ \(x\) is less than \(1 . "\)

a. Check to determine whether \((-3,10)\) satisfies the compound inequality \(-54\) in the rectangular coordinate system.

The double inequality \(4<3 x+5 \leq 15\) is equivalent to \(4<3 x+5,3 x+5 \leq 15\)

\(f(x)=|6 x-2|\) is called an absolute value _________.

Fill in the blanks. To _____ an inequality means to find all values of the variable that make the inequality true.

Fill in the blanks. a. When solving a compound inequality containing the word and, the solution set is the _________ of the solution sets of the inequalities. b. When solving a compound inequality containing the word or, the solution set is the _________ of the solution sets of the inequalities.

a. To graph the inequality \(y>3 x-1\), we begin by graphing the boundary line \(y=3 x-1 .\) What is the slope \(m\) of the line? What is its \(y\) -intercept? b. To graph the inequality \(2 x+3 y \leq-6,\) we begin by graphing the boundary line \(2 x+3 y=-6 .\) What are its \(x\) - and \(y\) -intercepts?

Which of the following are inequalities? $$ 6-x=8 \quad 5+a \quad 7 t-5>4 \quad \frac{x}{2} \leq-1 $$

Two absolute value expressions are equal when the expressions within the absolute value bars are equal to or _______ of each other.

Fill in the blanks. When multiplying or dividing all three parts of a double inequality by a negative number, the direction of both inequality symbols must be _______.

Perform each step listed below on the inequality \(4>-2\) and give the resulting true inequality. a. Add 2 to both sides. b. Subtract 4 from both sides. c. Multiply both sides by 4. d. Divide both sides by \(-2\)

Use a check to determine whether \(-3\) is a solution of the compound inequality. a. \(\frac{x}{3}+1 \geq 0\) and \(2 x-3<-10\) b. \(2 x \leq 0\) or \(-3 x<-5\)

Tell whether the graph of each inequality includes the boundary line. In each case, would the boundary be a solid or a dashed line? a. \(y<3 x-1\) b. \(2 x+3 y \geq-6\) c. \(y \leq-10\) d. \(x>1\)

Use a check to determine whether each number is a solution of \(3 x+6 \leq 6\). a. 0 b. \(\frac{2}{3}\) c. \(-10\) d. 1.5

Determine whether \(-3\) is a solution of the given equation or inequality. a. \(|x-1|=4\) b. \(|x-1|>4\) c. \(|x-1| \leq 4\) d. \(|5-x|=|x+12|\)

Use a check to determine whether \(-3\) is a solution of the double linear inequality. a. \(-1<-3 x+4<12\) b. \(-1<-3 x+4<14\)

Graph the solution set of each system of inequalities. See Example 1. $$\left\\{\begin{array}{l}3 x+y \leq 1 \\\\-x+2 y \geq 6\end{array}\right.$$

For each absolute value equation, write an equivalent compound equation. a. \(|x-7|=8\) is equivalent to \(x-7=\quad\) or \(\quad x-7=\) b. \(|x+10|=|x-3|\) is equivalent to \(x+10=\) or \(x+10=\)

Graph each inequality. $$ y>x+1 $$

Insert the correct symbol, \(<, \leq,>,\) or \(\geq,\) in each blank. a. As many as 16 people were seriously injured: The number of people seriously injured \(\square\) 16. b. There were no fewer than 8 references to taxes in the speech: The number of tax references \(\square\) 8. c. The weight \(w\) of the roast is at most 8 pounds: \(w\) \(\square\) 8. d. The temperature \(t\) exceeded \(100^{\circ}: t \square 100\)

For each absolute value equation or inequality, write an equivalent compound equation or inequality. a. \(|x|=8\) b. \(|x| \geq 8\) c. \(|x| \leq 8\) d. \(|5 x-1|=|x+3|\)

Graph each inequality. $$ y \geq-\frac{3}{2} x+1 $$

Perform the necessary steps to isolate the absolute value expression on one side of the equation. Do not solve. a. \(|3 x+2|-7=-5\) b. \(6+2|5 x-19| \leq 40\)

Graph each inequality. $$ y<\frac{x}{3}-1 $$

Determine the solution set of each absolute value equation or inequality by inspection. (No work is necessary.) Your answer should be either all real numbers or no solution. a. \(|7 x+6|=-8\) b. \(|7 x+6| \leq-8\) c. \(|7 x+6| \geq-8\)

Graph the solution set of each system of inequalities. See Example 2 . $$\left\\{\begin{array}{l}2 x+3 y \leq 6 \\\3 x+y \leq 1 \\\x \leq 0\end{array}\right.$$

What set is represented by the interval notation \((-\infty, \infty) ?\) Graph it.

Complete the solution to solve the inequality. Fill in the blank: If \(-10>x\) then \(x \square-10 .\)

Graph each inequality. $$ 2 x+y \leq 6 $$

Write the inequality \(10>|16 x-3|\) in an equivalent form with the absolute value expression on the left side.

Graph the solution set of each system of inequalities. See Example 2 . $$\left\\{\begin{array}{l}2 x+y \leq 2 \\\y \geq x \\\x \geq 0\end{array}\right.$$

a. Graph: \((-\infty, 2) \cup[3, \infty)\) b. Graph: \((-\infty, 3) \cap[-2, \infty)\)