Solving Equations and Inequalities

Solve each inequality. Graph the solution set on a number line.

$\left|3w+2\right|\le 5$

Solve each inequality. Graph the solution set on a number line.

$\left|7x\right|+4<0$

Solve each inequality. Graph the solution set on a number line.

41. $\left|n\right|\ge n$

Solve each inequality. Graph the solution set on a number line.

$\left|n\right|\le n$

Solve each inequality. Graph the solution set on a number line.

43. $\left|2n-7\right|\le 0$

Solve each inequality. Graph the solution set on a number line.

$\left|n-3\right|<n$

BETTA FISH

A Siamese Fighting Fish, also known as a Betta fish, is one of the most recognized and colourful fish kept as a pet. Using the information at the left, write a compound inequality to describe the acceptable range of water pH levels for a male Betta.

Betta Fish

Adult Male size: 3 inches

Water pH: $6.8-7.4$

Temperature: $75-86{}^{\circ }\text{F}$

Diet: omnivore, prefers live foods

Tank Level: top dweller

Difficulty of care: easy to intermediate

Life span: 2-3 years

A Siamese Fighting Fish, also known as a Betta fish, is one of the most recognized and colourful fish kept as a pet. Using the information at the left, write a compound inequality to describe the acceptable range of water pH levels for a male Betta.

Betta Fish

Adult Male size: 3 inches

Water pH: $6.8-7.4$

Temperature: $75-86{}^{\circ }\text{F}$ 

Diet: omnivore, prefers live foods

Tank Level: top dweller

Difficulty of care: easy to intermediate

Life span: 2-3 years

For exercises 46 and 47, use the following information.

On some interstate highways, the maximum speed a car may drive is 65 miles per hour. A tractor-trailer may not drive more than 55 miles per hour. The minimum speed for all vehicles is 45 miles per hour.

46. Write an inequality to represent the allowable speed for a car on an interstate highway.

On some interstate highways, the maximum speed a car may drive is 65 miles per hour. A tractor-trailer may not drive more than 55 miles per hour. The minimum speed for all vehicles is 45 miles per hour.

Write an inequality to represent the speed at which a tractor-trailer may travel on an interstate highway.

Hypothermia and hyperthermia are similar words but have opposite meanings. Hypothermia is defined as a lowered body temperature. Hyperthermia means extremely high body temperature, both conditions are potentially dangerous and occur when a person’s body temperature fluctuates by more than $\mathbf{8}\mathbf{°}$ from the normal body temperature of $\mathbf{98}\mathbf{.}\mathbf{6}{}^{\mathbf{\circ }}\mathbf{F}$. Write and solve an absolute value inequality to describe body temperatures that are considered potentially dangerous.

For exercises 49 and 50, use the following information.

The U.S. Postal Service defines an oversized package as one for which the length L of its longest side plus the distance D around its thickest part is more than 108 inches and less than or equal to 130 inches.

Write a compound inequality to describe this situation.

For exercises 49 and 50, use the following information.

The U.S. Postal Service defines an oversized package as one for which the length $\mathbf{L}$ of its longest side plus the distance $\mathbf{D}$around its thickest part is more than 108 inches and less than or equal to 130 inches.

If the distance around the thickest part of a package you want to mail is 24 inches, describe the range of lengths that would classify your package as oversized.

The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.

Write three inequalities to express the relationships among the sides of $\mathbf{\Delta ABC}$.

For exercises 51 and 52, use the following information. 

The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side.

51. Write a compound inequality to describe the range of possible measures for side $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Assume that $\mathbf{a}>\mathbf{b}>\mathbf{c}$. (Hint: Solve each inequality you wrote in exercise 51 for c)

Graph each set on a number line.

     a.$\mathbf{-}\mathbf{2}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{4}$

     b.$\mathbf{x}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{or}\mathbf{ }\mathbf{x}\mathbf{>}\mathbf{3}$

     c.$\left(\mathbf{-}\mathbf{2}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{4}\right)\mathbf{and}\left(\mathbf{x}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{orx}\mathbf{>}\mathbf{3}\right)$ (Hint: This is the intersection of the graphs in part a and part b.)

   d.Solve $\mathbf{3}\mathbf{<}\left|\mathbf{x}\mathbf{+}\mathbf{2}\right|\le \mathbf{8}$. Explain your reasoning and graph the solution set.

Explain how compound inequalities are used in medicine.

Solve $\left|\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{11}\right|\mathbf{>}\mathbf{1}$ for $\mathbf{x}$.

If , $\mathbf{5}\mathbf{<}\mathbf{a}\mathbf{<}\mathbf{7}\mathbf{<}\mathbf{b}\mathbf{<}\mathbf{14}$ then which of the following best defines $\frac{a}{b}$.?

For Exercises 57-60, use the following information.

You can use the operators in the LOGIVC menu on the Ti-83 Plus to graph compound and absolute value inequalities. To display the LOGIC menu, press $\boxed{2\text{nd}}\boxed{\text{TEST}}\boxed{}$.

Clear the Y= list. Enter $\left(5x+2>12\right)$ and $\left(3x-8<1\right)$as Y1. With your calculator in DOT mode and using the standard viewing window, press $\boxed{\text{GRAPH}}$. Make a sketch of the graph displayed.

For Exercises 57-60, use the following information.

You can use the operators in the LOGIVC menu on the Ti-83 Plus to graph compound and absolute value inequalities. To display the LOGIC menu, press  $\boxed{2\text{nd}}\boxed{\text{TEST}}\boxed{}$.

Use the TRACE function, investigate the graph. Based on your investigation, what inequality is graphed?

For Exercises 57-60, use the following information.

You can use the operators in the LOGIC menu on the Ti-83 Plus to graph compound and absolute value inequalities. To display the LOGIC menu, press $\boxed{2\text{nd}}\boxed{\text{TEST}}\boxed{}$.

Write the expression you would enter for Y1 to find the solution set of the compound inequality $5x+2\ge 3$or $5x+2\le -3$. Then use the graphing calculator to find the solution set.

For Exercises 57-60, use the following information.

You can use the operators in the LOGIC menu on the Ti-83 Plus to graph compound and absolute value inequalities. To display the LOGIC menu, press $\boxed{2\text{nd}}\boxed{\text{TEST}}\boxed{}$.

A graphing calculator can also be used to solve absolute value inequalities. Write the expression you would enter for Y1 to find the solution set of the inequality$\left|2x-6\right|>10$. Then use the graphing calculator to find the solution set. (Hint: the absolute value operator is item 1 on the MATH NUM menu.)

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

$\mathbf{7}\mathbf{x}\mathbf{+}\mathbf{11}>\mathbf{9}\mathbf{x}\mathbf{+}\mathbf{3}$

set on a number line.Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution 

63. $\mathbf{3}\mathbf{n}\mathbf{+}\mathbf{4}\left(\mathbf{n}\mathbf{+}\mathbf{3}\right)<\mathbf{5}\left(\mathbf{n}\mathbf{+}\mathbf{2}\right)$

To get a chance to win a car, you must guess the number of keys in a jar to within 5 of the actual number. Those who are within this range are given a key to try in the ignition of the car. Suppose there are 587 keys in the jar. Write and solve an equation to determine the highest and lowest guesses that will give contestants a chance to win the car.

Solve each equation. Check your solutions.

$\mathbf{5}\left|\mathbf{x}\mathbf{-}\mathbf{3}\right|\mathbf{=}\mathbf{65}$

Solve each equation. Check your solutions.

66. $\left|\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{7}\right|\mathbf{=}\mathbf{15}$

Solve each equation. Check your solutions.

67. $\left|\mathbf{8}\mathbf{c}\mathbf{+}\mathbf{7}\right|\mathbf{=}\mathbf{-}\mathbf{4}$

Name the property illustrated by each statement.

If $\mathbf{3}\mathbf{x}\mathbf{=}\mathbf{10}$, then $\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{7}\mathbf{=}\mathbf{10}\mathbf{+}\mathbf{7}$.

Name the property illustrated by each statement.

69. If $\mathbf{-}\mathbf{5}\mathbf{=}\mathbf{4}\mathbf{y}\mathbf{-}\mathbf{8}$, then $\mathbf{4}\mathbf{y}\mathbf{-}\mathbf{8}\mathbf{=}\mathbf{-}\mathbf{5}$.

Name the property illustrated by each statement.

70. If $\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{5}\mathbf{=}\mathbf{9}$and $\mathbf{9}\mathbf{=}\mathbf{6}\mathbf{x}\mathbf{+}\mathbf{1}$, then$\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{5}\mathbf{=}\mathbf{6}\mathbf{x}\mathbf{+}\mathbf{1}$

Simplify each expression.

$\mathbf{6}\mathbf{a}\mathbf{-}\mathbf{2}\mathbf{b}\mathbf{-}\mathbf{3}\mathbf{a}\mathbf{+}\mathbf{9}\mathbf{b}$

 Simplify each expression.

 $\mathbf{-}\mathbf{2}\left(\mathbf{m}\mathbf{-}\mathbf{4}\mathbf{n}\right)\mathbf{-}\mathbf{3}\left(\mathbf{5}\mathbf{n}\mathbf{+}\mathbf{6}\right)$

Find the value of each expression.

$\mathbf{6}\left(\mathbf{5}\mathbf{-}\mathbf{8}\right)\mathbf{÷}\mathbf{9}\mathbf{+}\mathbf{4}$

Find the value of each expression.

74. ${\left(\mathbf{3}\mathbf{+}\mathbf{7}\right)}^{\mathbf{2}}\mathbf{-}\mathbf{16}\mathbf{÷}\mathbf{2}$

Find the value of each expression.

$\frac{\mathbf{7}\left(\mathbf{1}\mathbf{-}\mathbf{4}\right)}{\mathbf{8}\mathbf{-}\mathbf{5}}$

Choose the term from the list above that best matches each example.

Choose the term from the list above that best matches each example.

Choose the term from the list above that best matches each example.

Choose the term from the list above that best matches each example.

$\mathbf{35}\mathbf{x}\mathbf{+}\mathbf{56}\mathbf{=}\mathbf{7}\left(\mathbf{5}\mathbf{x}\mathbf{+}\mathbf{8}\right)$

Choose the term from the list above that best matches each example.

 $\mathbf{ab}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{ab}\mathbf{+}\mathbf{1}$

Choose the term from the list above that best matches with each example.

6. If $\mathbf{2}\mathbf{x}\mathbf{=}\mathbf{3}\mathbf{y}\mathbf{-}\mathbf{4}$, $\mathbf{3}\mathbf{y}\mathbf{-}\mathbf{4}\mathbf{=}\mathbf{7}$, then $\mathbf{2}\mathbf{x}\mathbf{=}\mathbf{7}$.

Choose the term from the list above that best matches with each example.

$\mathbf{4}\left(\mathbf{0}\mathbf{.}\mathbf{25}\right)\mathbf{=}\mathbf{1}$

Choose the term from the list above that best matches with each example.

$\mathbf{2}\mathbf{p}\mathbf{+}\left(\mathbf{4}\mathbf{+}\mathbf{9}\mathbf{r}\right)\mathbf{=}\left(\mathbf{2}\mathbf{p}\mathbf{+}\mathbf{4}\right)\mathbf{+}\mathbf{9}\mathbf{r}$

Choose the term from the list above that best matches with each example.

$\left|\mathbf{5}\mathbf{n}\right|$

Choose the term from the list above that best matches with each example.

10. $\mathbf{6}\mathbf{y}\mathbf{+}\mathbf{5}\mathbf{z}\mathbf{-}\mathbf{2}\left(\mathbf{x}\mathbf{+}\mathbf{y}\right)$

Find the value of each expression

11. $10+16÷4+8$

Find the value of each expression

12. $\left[21-\left(9-2\right)\right]÷2$