Chapter 8

Explain how solving equations with variables on each side is like solving equations with variables on just one side. Include examples of both types of equations and an explanation of how they are alike and how they are different.

OPEN ENDED Give an example of an equation that has no solution and an equation that is an identity.

The formula \(F=\frac{9}{5} C+32\) is used to find the Fahrenheit temperature when a Celsius temperature is known. For what value are the Celsius and Fahrenheit temperatures the same? A. \(-72^{\circ}\) B. \(-40^{\circ}\) C. \(0^{\circ}\) D. \(32^{\circ}\)

Solve each inequality. Check your solution. Then graph the solution on a number line. $$\frac{m}{-7} \leq 1.2$$

Tim is taking the train to Seattle to visit his grandparents. He was given \(\$ 5.00\) to spend on snacks and reading material. Granola bars cost \(\$ 0.75\) each and a newspaper is \(\$ 1.25 .\) If Tim buys a newspaper, how many granola bars can he get?

CHALLENGE An apple costs the same as 2 oranges. Together, an orange and a banana cost \(10 \notin\) more than an apple. Two oranges cost \(15 \notin\) more than a banana. What is the cost for one of each fruit?

Two weeks ago the sewing club had 1 less than 3 times their average attendance. Last week they had 3 more than their average attendance. If the attendance for both weeks were equal, what is the average attendance of the sewing club? F. 1 G. 3 H. 2 J. 4

Write an inequality that can be solved using the Division Property of Inequality, where the inequality symbol is not reversed.

Solve each inequality. Check your solution. $$1+y \leq 2.4$$

Write a multi-step inequality that can be solved by first adding 3 to each side.

The product of an integer and \(-7\) is less than \(-84 .\) Find the least integer that meets this condition.

Solve each inequality. Check your solution. $$2.9

Assume that \(k\) is an integer. Solve the inequality \(10-2|k|>4\).

NUMBER SENSE Three times the sum of three consecutive integers \(x\) \(x+1,\) and \(x+2,\) is 72. What are the integers?

There are more than \(32,150\) high school girls basketball and track programs in the United States. If there are \(15,089\) girls track programs, write and solve an inequality to determine the number of girls basketball programs.

Find the true statement. \(\cdot\) A line of fit is close to most of the data points. \(\cdot\) A line of fit describes the exact coordinates of each point in the data set. \(\cdot\) A line of fit always has a positive slope.

Solve each inequality. Check your solution. $$f-4 \geq 1.4$$

Writing in Math Why is the Distributive Property important for solving equations? Include in your answer a definition of the Distributive Property and a description of its use in solving equations.

Use the Internet or another source to find the state or national spending limits on certain government branches, organizations, or projects. Write an inequality to express one or more of these limits.

What equation represents the table of values? $$\begin{array}{|c|c|c|c|c|}\hline x & -4 & -8 & -12 & -16 \\\\\hline y & 6 & 8 & 10 & 12 \\\\\hline\end{array}$$

Solve each inequality. Check your solution. $$z-2>-3.8$$

Wes leaves downtown driving 55 miles per hour. Emma follows 1 hour later, driving 60 miles per hour. Which equation can be used to determine how long it is after Wes leaves that Emma will catch up? A \(55 x=60 x-1\) B \(60 x=55(x-1)\) C \(55 x=60 x\) D \(55 x=60(x-1)\)

Provide a counterexample to the statement, "All numbers less than 0 are negative integers."

Use the Distributive Property to rewriteh expression as an equivalent algebraic expression. \(4(x-8)\)

Solve each inequality. Check your solution. $$b-\frac{3}{4}<2 \frac{1}{2}$$

Sandra's scores on the first five science tests are shown in the table. Which inequality represents the score she must receive on the sixth test to have an average score of more than \(88 ?\) A \(s \geq 86\) \(\mathbf{B} s \leq 88\) \(C s<88\) \(\mathbf{D} s>86\) $$\begin{array}{|c|c|}\hline \text { Test } & \text { Score } \\\\\hline 1 & 85 \\\\\hline 2 & 84 \\\\\hline 3 & 90 \\ \hline 4 & 95 \\\\\hline 5 & 88 \\\\\hline\end{array}$$

Write four examples of inequalities, each using one of the symbols \(<,>, \leq,\) and \(\geq .\) Tell the meaning of each inequality.

Use the Distributive Property to rewriteh expression as an equivalent algebraic expression. \(2(1.2 c+14)\)

Isabel is putting water into a 20 -gallon fish tank using a 2-quart pitcher. How many pitchers of water will she need to fill the tank?

Solve each inequality. Check your solution. $$g-1 \frac{2}{3}>2 \frac{1}{6}$$

Solve each equation. Check your solution. $$4 x=2 x+5$$

Which number is NOT a possible length of the rectangle if the area is less than 36 square inches? CAN'T COPY THE GRAPH $$\mathbf{F} \quad 6$$ $$\mathbf{G} \quad 7$$ $$\mathbf{H} \quad 8$$ $$\mathbf{J} \quad 9$$

Graph the solutions for each compound inequality. a. \(y<-2\) or \(y>3\) (Hint: In a sentence, or means either part is true.) b. \(y \geq 0\) and \(y \leq 5\) (Hint: In a sentence, and means both parts must be true.)

Use the Distributive Property to rewriteh expression as an equivalent algebraic expression. \(\frac{1}{2}(n-9)\)

Solve each inequality. Check your solution. $$6 x<-27$$

Solve each equation. Check your solution. $$3 x+5=7-2 x$$

Solve each inequality. Check your solution. $$-4+x>23$$

How can inequalities help you describe relationships? Illustrate your answer with a real-world example that uses an inequality symbol and an explanation of the relationships described by the inequality.

Solve each inequality. Check your solution. $$-5 m \geq-15$$

Solve each equation. Check your solution. $$1.5 x+9=3 x-3$$

Solve each inequality. Check your solution. $$c+18 \leq-2$$

CHALLENGE Is it always, sometimes, or never true that \(x-1

Solve each inequality. Check your solution. $$8>\frac{q}{3}$$

Solve each inequality. Check your solution. $$6>n-10$$

Writing in Math How is solving an inequality similar to solving an equation?

Solve each inequality. Check your solution. $$\frac{n}{-4} \leq-11$$

Express each number in scientific notation. $$4,500,000$$

Solve each equation. Check your solution. $$2(3+x)=14$$

It takes Carolyn two hours to complete a cross-stitch pattern. Carolyn can spend no more than fourteen hours cross-stitching. Write an inequality that represents this situation and use it to determine whether Carolyn can complete 8 cross-stitch patterns.

Trevor has \(\$ 25\) to spend on a T-shirt and shorts for gym class. The shorts cost \(\$ 14\). Based on the inequality \(14+t \leq 25,\) where \(t\) represents the cost of the T-shirt, what is the most Trevor can spend on the T-shirt?