2.7 Linear Inequalities and Absolute Value Inequalities

Use interval notation to indicate all real numbers greater than or equal to $−2.$

Use interval notation to indicate all real numbers between and including $−3$ and $5.$

Write the interval expressing all real numbers less than or equal to $−1$ or greater than or equal to $1.$

Express all real numbers less than $−2$ or greater than or equal to 3 in interval notation.

Illustrate the addition property for inequalities by solving each of the following:

1. ⓐ $x - 15 < 4$
2. ⓑ $6 \geq x - 1$
3. ⓒ $x + 7 > 9$

Solve: $3 x −2 < 1.$

Illustrate the multiplication property for inequalities by solving each of the following:

1. ⓐ $3 x < 6$
2. ⓑ $−2 x - 1 \geq 5$
3. ⓒ $5 - x > 10$

Solve: $4 x + 7 \geq 2 x - 3.$

Solve the inequality: $13 - 7 x \geq 10 x - 4.$

Solve the inequality and write the answer using interval notation: $- x + 4 < \frac{1}{2} x + 1.$

Solve the following inequality and write the answer in interval notation: $- \frac{3}{4} x \geq - \frac{5}{8} + \frac{2}{3} x .$

Solve the inequality and write the answer in interval notation: $- \frac{5}{6} x \leq \frac{3}{4} + \frac{8}{3} x .$

Solve the compound inequality: $3 \leq 2 x + 2 < 6.$

Solve the compound inequality: $4 < 2 x - 8 \leq 10.$

Solve the compound inequality: $3 y < 4 - 5 y < 5 + 3 y .$

Describe all x-values within a distance of 3 from the number 2.

Solve $\left|x - 1\right| \leq 3$ .

When solving an inequality, explain what happened from Step 1 to Step 2:

$\begin{matrix}\text{Step 1} & - 2 x > 6 \\ \text{Step 2} & x < - 3\end{matrix}$

When writing our solution in interval notation, how do we represent all the real numbers?

Describe how to graph $y = \left|x - 3\right|$

Solve the inequality. Write your final answer in interval notation.

$3 x + 2 \geq 7 x - 1$

Solve the inequality. Write your final answer in interval notation.

$4 \left(x + 3\right) \geq 2 x - 1$

Solve the inequality. Write your final answer in interval notation.

$−5 \left(x - 1\right) + 3 > 3 x - 4 - 4 x$

Solve the inequality. Write your final answer in interval notation.

$\frac{x + 3}{8} - \frac{x + 5}{5} \geq \frac{3}{10}$

Solve the inequality involving absolute value. Write your final answer in interval notation.

$\left|x + 9\right| \geq −6$

Solve the inequality involving absolute value. Write your final answer in interval notation.

$\left|3 x - 1\right| > 11$

Solve the inequality involving absolute value. Write your final answer in interval notation.

$\left|x - 2\right| + 4 \geq 10$

Solve the inequality involving absolute value. Write your final answer in interval notation.

$\left|x - 7\right| < −4$

Solve the inequality involving absolute value. Write your final answer in interval notation.

$\left|\frac{x - 3}{4}\right| < 2$

Describe all the x-values within or including a distance of the given values.

Distance of 3 units from the number 9

Describe all the x-values within or including a distance of the given values.

Distance of 11 units from the number 1

Solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.

$3 x + 1 > 2 x - 5 > x - 7$

Solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.

$2 x - 5 < −11 \text{or } 5 x + 1 \geq 6$

Write the set in interval notation.

$\left\{\right. x \left|\right. −1 < x < 3 \left.\right\}$

Write the set in interval notation.

$\left\{\right. x \left|\right. x < 4 \left.\right\}$

Write the interval in set-builder notation.

$\left(- \infty , 6\right)$

Write the interval in set-builder notation.

$\left[−3 , 5\right)$

Solve $\left|3 x + 1\right| = \left|2 x + 3\right|$

$\frac{x - 5}{x + 7} \leq 0 ,$ $x \neq −7$

In chemistry the volume for a certain gas is given by $V = 20 T ,$ where V is measured in cc and T is temperature in ºC. If the temperature varies between 80ºC and 120ºC, find the set of volume values.

Find the x-intercept and the y-intercept without graphing.

$4 x - 3 y = 12$

Solve for y in terms of x, putting the equation in slope–intercept form.

$5 x = 3 y - 12$

Find the distance between the two points.

$\left(−2 , 5\right) \left(4 , −1\right)$

Find the distance between the two points.

Find the distance between the two points $\left(−71,432\right)$ and $(\text{511},\text{218})$ using your calculator, and round your answer to the nearest thousandth.

Find the coordinates of the midpoint of the line segment that joins the two given points.

$\left(−13 , 5\right)$ and $\left(17 , 18\right)$

Solve for $x .$

$3 \left(x + 2\right) - 10 = x + 4$

Solve for $x .$

$12 - 5 \left(x + 1\right) = 2 x - 5$

Solve for $x .$ State all x-values that are excluded from the solution set.

$\frac{x}{x^{2} - 9} + \frac{4}{x + 3} = \frac{3}{x^{2} - 9}$ $x \neq 3 , −3$

Find the equation of the line using the point-slope formula.

Passes through these two points: $\left(−2 , 1\right) , \left(4 , 2\right) .$

Find the equation of the line using the point-slope formula.

Passes through the point $\left(- 3 , 4\right)$ and is parallel to the graph $y = \frac{2}{3} x + 5.$

Write and solve an equation to answer each question.

The number of male fish in the tank is five more than three times the number of females. If the total number of fish is 73, how many of each sex are in the tank?

Write and solve an equation to answer each question.

A truck rental is $25 plus $.30/mi. Find out how many miles Ken traveled if his bill was $50.20.

Use the quadratic equation to solve.

$2 x^{2} + 3 x + 7 = 0$

Name the horizontal component and the vertical component.

$−2 - i$

Perform the operations indicated.

$\left(2 + 3 i\right) - \left(- 5 - 8 i\right)$

Perform the operations indicated.

$\sqrt{- 16} + 4 \sqrt{- 9}$

Perform the operations indicated.

$\left(3 - 5 i\right)^{2}$

Perform the operations indicated.

$\sqrt{- 2} \left(\sqrt{- 8} - \sqrt{5}\right)$

Perform the operations indicated.

$\frac{3 + 7 i}{i}$

Solve the quadratic equation by factoring.

$3 x^{2} + 18 x + 15 = 0$

Solve the quadratic equation by factoring.

$7 x^{2} - 9 x = 0$

Solve the quadratic equation by using the square-root property.

$\left(x - 4\right)^{2} = 36$

Solve the quadratic equation by completing the square.

$4 x^{2} + 2 x - 1 = 0$

Solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution.

$15 x^{2} - x - 2 = 0$

Solve the quadratic equation by the method of your choice.

$x^{2} = 10 x + 3$

Solve the equations.

$x^{\frac{1}{2}} - 4 x^{\frac{1}{4}} = 0$

Solve the equations.

$3 x^{5} - 6 x^{3} = 0$

Solve the equations.

$\sqrt{3 x + 7} + \sqrt{x + 2} = 1$

Solve the equations.

$\left|2 x + 3\right| - 5 = 9$

Solve the inequality. Write your final answer in interval notation.

$- 2 x + 5 > x - 7$

Solve the inequality. Write your final answer in interval notation.

$\left|3 x + 2\right| + 1 \leq 9$

Solve the inequality. Write your final answer in interval notation.

$\left|x - 3\right| < −4$

Solve the compound inequality. Write your answer in interval notation.

$3 y < 1 - 2 y < 5 + y$

Write the interval notation for the set of numbers represented by $\left\{x \left|\right. x \leq 9\right\} .$

Solve for $x$: $3 \left(2 x - 5\right) - 3 \left(x - 7\right) = 2 x - 9.$

Solve for x: $\frac{5}{x + 4} = 4 + \frac{3}{x - 2} .$

Solve for x. Write the answer in simplest radical form.

$\frac{x^{2}}{3} - x = - \frac{1}{2}$

Solve: $\left|2 x + 3\right| < 5.$

Find the equation of the line with the given information.

Passes through the points $\left(- 4 , 2\right)$ and $\left(5 , −3\right) .$

Find the equation of the line with the given information.

Passes through the point $\left(2 , 1\right)$ and is perpendicular to $y = - \frac{2}{5} x + 3.$

Find the equation of the line with the given information.

Simplify: $\sqrt{−4} + 3 \sqrt{−16} .$

Find the equation of the line with the given information.

Divide: $\frac{4 - i}{2 + 3 i} .$

Find the equation of the line with the given information.

Solve: $\left(3 x - 1\right)^{2} - 1 = 24.$

Find the equation of the line with the given information.

Solve: $4 x^{2} - 4 x - 1 = 0$

Find the equation of the line with the given information.

Solve: $2 + \sqrt{12 - 2 x} = x$

Find the real solutions of each equation by factoring.

$2 x^{3} - x^{2} - 8 x + 4 = 0$