Solving Equations and Inequalities

Solve the equation. Check the solutions.

$|3x-7|-5=-3$

Solve the equation. Check the solutions.

$4|3t+8|=16t$

Solve the equation. Check the solutions.

$|15+m|=-2m+3$

COFFEE

Some say that to brew an excellent cup of coffee, you must have a brewing temperature of $200{}^{\circ }F$, plus minus five degrees. Write and solve an equation describing the maximum and minimum brewing temperatures for an excellent cup of coffee.

A machine is used to fill each of several bags with 16 ounces of sugar. After the bags are filled, another machine weighs them. If the bag weighs $0.3$ ounce more or less than the desired weight, the bag is rejected. Write an equation to find the heaviest and lightest bag the machine will approve.

The atmosphere of Earth is divided into four layers based on temperature variations. The troposphere is the layer closest to the planet. The average upper boundary of the layer is about 13 kilometres above Earth’s surface. This height varies with latitude and with the seasons by as much as 5 kilometres. Write and solve an equation describing the maximum and minimum heights of the upper bound of the troposphere.

For the following statement, determine whether the statement is always, sometimes or never true. Explain your reasoning.

 If $a$and $b,c$ real numbers, then $c|a+b|=\left|ca\right|+\left|cb\right|$.

Answer the question that was posed at the beginning of the lesson.

How can an absolute value equation describe the magnitude of an earthquake?

Include the following in your answer:

• a verbal and graphical explanation of how $\left|E-6.1\right|=0.3$ describes the possible extremes in the variation of the earthquake’s magnitude, and
• an equation to describe the extremes for a different magnitude.

Which of the graphs below represents the solution set for $|x-3|-4=0$.

Find the value of $-|-9|-\left|4\right|-3|5-7|$.

$A.-19 B.-11 C.-7 D.11$

For Exercises 55-58, consider the equation $\left|x+1\right|+2=\left|x+4\right|$.

55. To solve this equation we must consider the case where $x+4\ge 0$ and the case where $x+4<0$ . Write the equations for each of these cases.

For Exercises 55-58, consider the equation $\left|x+1\right|+2=\left|x+4\right|$.

Notice that each equation you wrote in exercise 55 has two cases. For each equation, write two other equations taking into consideration the case where $x+1\ge 0$ and the case where $x+1<0$ .

For Exercises 55-58, consider the equation $\left|x+1\right|+2=\left|x+4\right|$.

Solve each equation that you wrote in Exercise 56. Then, check each solution in the original equation, $\left|x+1\right|+2=\left|x+4\right|$. What are the solution(s) to this absolute value equation?

For Exercises 55-58, consider the equation $\left|x+1\right|+2=\left|x+4\right|$.

58. MAKE A CONECTURE

For equations with one set of absolute value symbols, two cases must be considered. For an equation with two sets of absolute value symbols, four cases must be considered. How many cases must be considered for an equation containing three sets of absolute value symbols?

Write an algebraic expression to represent the verbal expression.

Twice the difference of a number and 11.

Write an algebraic expression to represent the verbal expression.

The product of the square of a number and 5.

Solve the equation and check the solution.

$3x+6=22$

Solve the equation and check the solution.

$7p-4=3\left(4+5p\right)$

Solve the equation and check the solution.

$\frac{5}{7}y-3=\frac{3}{7}y+1$

Name the property illustrated by the following equation.

$\left(5+9\right)+13=5+(9+13)$

Name the property illustrated by the following equation.

$m\left(4-3\right)=m·4-m·3$

Name the property illustrated by the following equation.

$\left(\frac{1}{4}\right)4=1$

Name the property illustrated by the following equation.

$5x+0=5x$

Determine whether each statement is true or false. If false, give a counterexample.

Every real number is a rational number. 

Determine whether each statement is true or false. If false, give a counter example.

Every natural number is an integer. 

Determine whether each statement is true or false. If false, give a counterexample.

Every irrational number is a real number. 

Determine whether each statement is true or false. If false, give a counter example.

Every rational number is an integer. 

For the following figure, the formula for the area A of a triangle is $A=\frac{1}{2}bh$ . Where b is the measure of the base and h is the measure of height. Write an expression to represent the area of the triangle.

For the following figure, the formula for the area A of a triangle is $A=\frac{1}{2}bh$ . Where b is the measure of the base and h is the measure of height. Write the area of the triangle when x=23.

Solve the equation.

$14y-3=25$

Solve the equation.

$4.2x+6.4=40$

Solve the equation.

$7w+2=3w-6$

Solve the equation.

$2\left(a-1\right)=8a-6$

Solve the equation.

$48+5y=96-3y$

Solve the equation.

$\frac{2x+3}{5}=\frac{3}{10}$

Explain why it is not necessary to state a division property for inequalities.

Write an inequality using the $>$ symbol whose solution set is graphed below.

Write an inequality for which solution set is the empty set.

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

$a+2<3.5$

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

5. $5\ge 3x$

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

 $11-c\le 8$

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

$4y+7>31$

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

8. $2w+19<5$

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

 $-0.6p<-9$

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

10. $\frac{n}{12}+15\le 13$

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

$\frac{5z+2}{4}<\frac{5z}{4}+2$

Define a variable and write an inequality for each problem. Then solve.

12. The product of 12 and a number is greater than 36. 

Define a variable and write an inequality for each problem. Then solve.

Three less than twice a number is at most 5.

The final grade for a class is calculated by taking $75\%$ of the average test score and adding $25\%$ of the score on the final exam. If all scores are out of 100 and a student has a 76 test average what score does the student need to make on the final exam to have a final grade of at least 80?

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

15. $n+4\ge -7$