Rationals

Problem 301

Simplify $\frac{2 x^{2}}{10 x^{3}-2 x^{2}}$ , $\frac{1}{10 x^{3}-1}$ , $\frac{1}{5 x-1}$ , $\frac{1}{10 x^{3}}$ .

See Solution

Problem 302

Identify which of the following expressions equals 1 and which equals -1:
$\frac{x+3}{3+x} \quad \frac{3-x}{x-3}$

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Problem 303

19. Find the exact value of $3-2 . \dot{9}$ .

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Problem 304

1. What is the rational number represented by letter $B$ .
1. $\qquad$ A) -1.1 B) 0.1 C) -0.1 D) 1.1

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Problem 305

A) -1.1 B) 0.1
2. Which number is a not a rational number?
2. $\qquad$ C) -0.1 A) $\sqrt{16}$ B) $\pi$ C) -5 D) 2
3. What number is equivlant to $\frac{-2}{3}$ ?
3. $\qquad$ A) $\frac{2}{3}$ B) $\frac{-3}{6}$ C) $\frac{-4}{-6}$ D) $\frac{-4}{6}$
4. $\qquad$
4. Evaluate: $\frac{-2}{5}+\left(\frac{3}{5}\right)$ A) $\frac{1}{10}$ B) $\frac{1}{5}$ C) $\frac{-1}{5}$ D) $\frac{-6}{25}$ 5.

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Problem 306

Pary 2: Long Answer Questions. 24 marks Answer ALL questions in the space provided. Show ALL working to receive FULL credit.
1. Place and label these rational numbers on the number line below: (5 marks) $-4.124, \frac{-3}{4}, 3 \frac{1}{2}, \frac{-11}{8}, 0 . \overline{9}$

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Problem 307

Answer ALL questions in the space provided. Show ALL working to receive FULL credit
1. Place and label these rational numbers on the number line below: $-4.124, \frac{-3}{4}, 3 \frac{1}{2}, \frac{-11}{8}, 0 . \overline{9}$ (5 marks)

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Problem 308

Calculate the value of $333 \sqrt{777}$ .

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Problem 309

What is the expression for Tran's gas cost per mile if he pays \ $84 to drive$ m$ miles?

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Problem 310

Simplify the expression $\left(\frac{a+1}{a-1}+\frac{a-1}{a+1}\right) \div\left(\frac{a+1}{a-1}-\frac{a-1}{a+1}\right)$ .

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Problem 311

Solve for $x$ in the equation $\frac{x}{x+3}=\frac{5}{4}$ .

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Problem 312

Solve the equation: $\frac{4}{x-4}=\frac{-4}{x+3}+\frac{12}{(x-4)(x+3)}$

See Solution

Problem 313

Solve for $x$ in the equation $\frac{3}{x}+3=\frac{3}{5 x}+\frac{31}{10}$ .

See Solution

Problem 314

Solve the inequality: $\frac{x+3}{x-1}>2$

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Problem 315

\begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|l|}{ Transformez les chiffres périodiques suivants en fraction: } \\ \hline $0.6666 \ldots .$ . & $1.2222 \ldots \ldots .$ . & $1.333 \ldots \ldots$ \\ \hline & & \\ \hline $0.45454545 \ldots \ldots$ . & 0.846153846153846153 & \\ \hline & & \\ \hline \end{tabular}

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Problem 316

\text{Write the recurring decimal } 99.9\overline{9} \text{ as a fraction in its lowest terms.}

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Problem 317

Solve for $x$ in the equation $gc=\frac{x+y}{-x}$ .

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Problem 318

Find numbers to exclude from the domain of these rational expressions: 1. $\frac{7}{x-3}$ , 2. $\frac{13}{x+9}$ , 3. $\frac{x+5}{x^{2}-25}$ , 4. $\frac{x+7}{x^{2}-49}$ , 5. $\frac{x-1}{x^{2}+11 x+10}$ , 6. $\frac{x-3}{x^{2}+4 x-45}$ .

See Solution

Problem 319

Simplify the expressions and find excluded values: 7. $\frac{3 x-9}{x^{2}-6 x+9}$ , 8. $\frac{4 x-8}{x^{2}-4 x+4}$ .

See Solution

Problem 320

Simplify the expression: $\frac{6 x+9}{3 x-15} \cdot \frac{x-5}{4 x+6}$ .

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Problem 321

Simplify the expression: $\frac{\frac{x}{3}-1}{x-3}$ .

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Problem 322

Add the fractions $\frac{8}{x-2}$ and $\frac{2}{x-3}$ .

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Problem 323

A boat takes $2 \frac{1}{2}$ hours and a plane takes $\frac{1}{2}$ hour. How much faster is the plane than the boat?

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Problem 324

Identify the mistake in $\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b}$ and correct the right side of the equation.

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Problem 325

Find the volume of ethanol that has the same mass as 0.950 L of mercury, given their densities: $13.546 \mathrm{~g/mL}$ for mercury and $0.789 \mathrm{~g/mL}$ for ethanol.

See Solution

Problem 326

Convert 0.6 to a simplified fraction. What is the lowest terms fraction for $0.6$ ?

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Problem 327

Multiply and simplify: $\left(-\frac{5}{7}\right)\left(-\frac{14}{15}\right) =$ (Reduce to lowest terms.)

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Problem 328

Multiply and simplify: $\frac{2}{11} \cdot \frac{2}{7}$ . Provide your answer as an integer or simplified fraction.

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Problem 329

Solve for $x$ : $\frac{\sqrt{x+3}}{x+3}=1$ . Options: no solution, $x=-2$ , $x=-3$ , $x=-3$ and $x=-2$ .

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Problem 330

Find the function $(s+t)(x)$ for $s(x)=\frac{x-2}{x^{2}-25}$ and $t(x)=\frac{x-5}{2-x}$ . Write the domain in interval notation.

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Problem 331

Find $(s \cdot t)(x)$ for $s(x) = \frac{x-3}{x^{2}-36}$ and $t(x) = \frac{x-6}{3-x}$ . State the domain in interval notation.

See Solution

Problem 332

Find $(s+t)(x)$ for $s(x)=\frac{x-3}{x^2-36}$ and $t(x)=\frac{x-6}{3-x}$ ; state the domain in interval notation.

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Problem 333

Find $(f \circ g)(x)$ and its domain in interval notation for $f(x)=\frac{x}{x-1}$ and $g(x)=\frac{13}{x^{2}-36}$ .

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Problem 334

Evaluate $(f \circ g)(x)$ and find its domain in interval notation, where $f(x)=\frac{x}{x-1}$ and $g(x)=\frac{13}{x^{2}-36}$ .

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Problem 335

Find $(f \circ g)(x)$ for $f(x)=\frac{x}{x+9}$ and $g(x)=\frac{8}{x^{2}-1}$ ; state the domain in interval notation.

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Problem 336

Find the domain of $(s \cdot t)(x)=\frac{-1}{x+6}$ , where $s(x)=\frac{x-3}{x^{2}-36}$ and $t(x)=\frac{x-6}{3-x}$ .

See Solution

Problem 337

Find the domain, range, inverse domain, and inverse range of the function $f(x)=\frac{5}{4 x+1}$ .

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Problem 338

Find the inverse of the one-to-one function $f(x)=\frac{2}{5 x+3}$ .

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Problem 339

Find the inverse of the one-to-one function $f(x)=\frac{6 x+7}{3 x+6}$ .

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Problem 340

Express the number as a ratio of integers. $2.53 \overline{38}$

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Problem 341

If 15 boys finish work in 60 days, how many boys are needed for the same work in 20 days?

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Problem 342

Solve for $x$ : $\frac{x+1}{x}+\frac{3}{4}=\frac{-3}{x}$ . What is $x$ ?

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Problem 343

Solve $\frac{7}{x}=\frac{2}{3 x}-10$ for $x$ . What is $x$ ?

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Problem 344

Solve for $x$ in the equation $\frac{1}{x+4}=\frac{4}{x-4}$ . What is $x$ ?

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Problem 345

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+3}$ , $h \neq 0$ . Simplify your answer.

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Problem 346

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{2}{x}$ , where $h \neq 0$ .

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Problem 347

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{9}{x^{2}}$ , simplifying where $h \neq 0$ .

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Problem 348

Solve the inequality $\frac{3 x-5}{-3}>5$ and provide the solution set in interval and graph forms.

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Problem 349

Identify which numbers in the set $E=\{\sqrt{14}, \pi, \sqrt{14}+1, \pi+\frac{1}{14}\}$ are (a) Natural, (b) Integers, (c) Rational, (d) Irrational, (e) Real.

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Problem 350

Calculate the expression $4 x y^{-1}$ for $x=9$ and $y=-2$ .

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Problem 351

Simplify the expression: $\frac{(-5)^{3} x^{7}(y z)^{3}}{(4)^{2} x^{5} y^{7} z^{2}}$

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Problem 352

Identify values to exclude from the domain of $\frac{x}{x^{2}-4}$ . Options: $x=0$ , $x=-2$ , $x=-1$ , $x=2$ , None.

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Problem 353

Identify values to exclude from the domain of $\frac{x^{2}}{x^{2}+16}$ . Options: $x=5$ , $x=4$ , $x=0$ , $x=-4$ . Select all that apply.

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Problem 354

Find values to exclude from the domain of $\frac{x^{2}+7 x-11}{x^{3}-4 x}$ . Options: $x=2$ , $x=11$ , $x=-2$ , $x=0$ . Select all that apply.

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Problem 355

Find the domain of $x$ for the expression $\frac{4}{x-19}$ .

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Problem 356

Find the domain of $x$ in the expression $\frac{x}{x+79}$ .

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Problem 357

EXERCICE 1: 1) calculer : $\frac{1}{2}+\left(\frac{-5}{16}\right) \quad ; \quad \frac{-2}{3}+\frac{8}{5} \quad ; \quad \frac{-3}{5} \times \frac{10}{3} ; \frac{1}{6}+\frac{-2}{3}$ 2) calculer et simplifier les expressions suivantes: $\begin{array}{ll} A=\frac{9}{5}-\frac{9}{5} \times \frac{5}{2} & B=\frac{3}{4}-\frac{5}{4} \times \frac{1}{10}-\frac{7}{10} \\ C=\left(\frac{2}{3}-\frac{1}{2}\right)\left(\frac{2}{5}-\frac{3}{4}\right) & D=\frac{1-\frac{5}{7}}{7}-1 \end{array}$ 3) Simplifier l'expression suivante tel que a et b deux nombres rationnels non nuls $\left.E=\frac{a+b}{\frac{1}{a}+\frac{1}{b}}\right\}$

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Problem 358

Solve the compound inequality: $0.5 < \frac{y+2}{2} < 5.5$ and express the solution in interval notation.

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Problem 359

Solve the compound inequality $\frac{22}{8} < \frac{y+7}{4} < \frac{29}{8}$ and express your answer in interval notation. Use decimals.

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Problem 360

Solve the inequality: $\frac{3 x-7}{-3}>7$ . Provide the solution set in interval notation and graph form.

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Problem 361

Graph the solution set for the compound inequality: $\frac{22}{8}<\frac{y+7}{4}<\frac{29}{8}$ .

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Problem 362

Find the value of $\frac{7 \div(q)^{2} \cdot 2}{2 p} \cdot \frac{-p+6 q-r}{-q}$ for $p=4$ , $q=\frac{1}{2}$ , $r=2$ .

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Problem 363

Solve the compound inequality $\frac{16}{8}<\frac{y+3}{4}<\frac{25}{8}$ and express the solution in interval notation.

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Problem 364

Solve the compound inequality $\frac{16}{8}<\frac{y+3}{4}<\frac{25}{8}$ and express the solution in interval notation.

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Problem 365

Solve the compound inequality $\frac{12}{8}<\frac{y+8}{4}<\frac{20}{8}$ and give the answer in interval notation.

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Problem 366

Classify 7.9 as a member of the following sets: natural, integer, rational, or real. Select all that apply.

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Problem 367

Determine the group of sets for the number $-\frac{7}{10}$ . Options: A. real, rational B. real, irrational C. real, rational, natural D. rational, natural, integers E. irrational, natural.

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Problem 368

Identify the rational numbers from this list: A. $-\frac{8}{9}$ , B. $\sqrt{2}$ , C. $0 . \overline{25}$ , D. $\pi$ , E. $\frac{2}{1}$ , F. None.

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Problem 369

Find the y-intercept of $f(x)=\frac{x^{2}-4}{x+2}$ .

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Problem 370

Determine the oblique asymptote for the function $f(x)=\frac{x^{2}-4}{x+2}$ .

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Problem 371

Determine the vertical asymptote of $f(x)=\frac{x^{2}-4}{x+2}$ .

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Problem 372

Determine the horizontal asymptote for the function $f(x)=\frac{x^{2}-4}{x+2}$ .

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Problem 373

Find the holes in the rational function $f(x)=\frac{x^{2}-4}{x+2}$ . If none, state 'none'.

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Problem 374

Simplify: $\frac{x^{2}-3 x-4}{x^{2}+x} \div \frac{x-4}{x^{2}}$

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Problem 375

Simplify these expressions: 1) $\frac{x^{2}-3 x-4}{x^{2}+x} \div \frac{x-4}{x^{2}}$ 2) $\frac{x-1}{x+2}+\frac{x+1}{2 x+4}$

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Problem 376

Simplify $\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m/n + 1\right)$ .

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Problem 377

Distribute in the expression: $x\left(1-\frac{4}{x}\right)=$

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Problem 378

Find $x$ in the proportion $9: 14:: x: 56$ .

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Problem 379

Caroline bought 20 shares at $10.50$ each. After 10 months, they are worth $11.25$ each. What is her profit? A. \$225 B. \$210 C. \$15 D. \$10

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Problem 380

A patient needs 260 mg of acetaminophen. How many mL of Liquid Tylenol ( $325 \mathrm{mg} / 12.5 \mathrm{ml}$ ) is required? Choose the two correct fractions for dimensional analysis.

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Problem 381

A patient needs 260 mg of acetaminophen. How many mL of Liquid Tylenol ( $325 \mathrm{mg} / 12.5 \mathrm{ml}$ ) is required?

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Problem 382

Find the inverse function $f^{-1}(x)$ for $f(x)=\frac{4 x+6}{5 x+3}$ .

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Problem 383

Find the inverse function $f^{-1}(x)$ for $f(x)=\frac{2x+3}{x-5}$ .

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Problem 384

Substitute b = 2 and c = 4 into $\frac{b^{2}-2 c^{2}}{a+c-b}$ and express in terms of 'a'.

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Problem 385

Calculate $\frac{b c^{2}+a}{c}$ for $a = 12$ , $b = 9$ , and $c = 4$ .

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Problem 386

Solve the inequality $\frac{5}{2 p}+\frac{15}{6 p}-\frac{3+2 p}{p} \geq 3$ for $p > 0$ . What is $p$ ?

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Problem 387

$5,229 \div 9$

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Problem 388

$2,025 \div 22=$ $\square$ R $\square$ Submit

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Problem 389

Use a calculator to approximate the square root. Check to see that the approximation is reasonable. $\sqrt{72}$ $\sqrt{72} \approx$ $\square$ (Round to the nearest thousandth as needed.)

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Problem 390

stard (-10) (0) () (4) (16) MODONNIT? Name I CAN ADD + SUBTRACT RATIONAL NUMBERS Use your solutions to eliminate suspects and solve the mystery! The last 3 remaining are the solutions.
1. -1+ (-2)
2. 一:+(一) -16) dio 5. I CAN ADD + SUBTRACT RATIONAL NUMBERS -2+ (-3) 10. 1-2/1 1-2 -2-3 11. -- (-4) 12.-2+1 --(-3) 6. -19-(-2) -3+1 MHODON 쯤 + (-3) SUSPECT VICTIM WEAPON SNOW ANGEL JOE COOL -3 9/10 THE GRINCH -41/12 TOM TURKEY -6 1/12 SANTA CLAUS 17/21 JACK FROST -57/12 -1 23/36 LUCKY LEPRECHAUN -7/15 GROOVY GAL -47/15 PUMPKIN HEAD -13/15 SNOWFLAKE 1 13/30 LEAD PIPE -22/5 BRICK -2 1/2 CANDLESTICK -1928 POISON -1 1/20 WRENCH 2 11/12 72512 09209

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Problem 391

Lesson Summary - Use the properties of operations to add and subtract rational numbers more efficiently. For instance, $-5 \frac{2}{9}+3.7+5 \frac{2}{9}=\left(-5 \frac{2}{9}+5 \frac{2}{9}\right)+3.7=0+3.7=3.7$ . - The opposite of a sum is the sum of its opposites as shown in the examples that follow: $\begin{array}{l} -4 \frac{4}{7}=-4+\left(-\frac{4}{7}\right) \\ -(5+3)=-5+(-3) \end{array}$
Problem Set
Show all steps taken to rewrite each of the following as a single rational number.
1. $80+\left(-22 \frac{4}{15}\right)$
2. $10+\left(-3 \frac{3}{8}\right)$
3. $\frac{1}{5}+20.3-\left(-5 \frac{3}{5}\right)$
4. $\frac{11}{12}-(-10)-\frac{5}{6}$

See Solution

Problem 392

Simplify the expression $(\sqrt{6}+5)(\sqrt{6}-5)$ .

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Problem 393

If a group plays 4 games, how much more do they spend buying single games at \$4.40 each vs. 2-game packs at \$8.40?

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Problem 394

How many cm does the map use to show 1 km if the scale is $0.3: \frac{1}{5}$ ? A $\frac{2}{3}$ B $\frac{3}{5}$ C 0.06 D 1.5

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Problem 395

Calculate the following: (a) $1.600 \times 10^{-7} \times 2.1 \times 10^{3}$ , (b) $(1.33)^{3}$ , (c) $1.93 \times 2.651$ , (d) $4.4 / 2.200$ .

See Solution

Problem 396

Find the mass percent of $\mathrm{K}_{2} \mathrm{SO}_{4}$ in a mixture that is $45.85\% \mathrm{SO}_{4}$ by mass.

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Problem 397

Evaluate $5\left[13-\left(3^{2}+2^{2}\right)\right]$ and find which is NOT equivalent to $2 \cdot 3 \frac{5}{6}+2 \cdot 12+2 \cdot 1 \frac{1}{6}$ .

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Problem 398

Is $\frac{\sqrt{10}}{5}$ rational? Is $\sqrt{18} - 3$ rational? Can two irrational numbers sum/quotient to rational? Explain.

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Problem 399

Simplify the expression $\frac{a^{4} \cdot b^{2}}{a b^{3}}$ .

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Problem 400

Rationalize the denominator of $\frac{4}{\sqrt{x}+1}$ .

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1 2 3 4 5 6 7 ...9

Rationals

Problem 301

Simplify 2x210x3−2x2\frac{2 x^{2}}{10 x^{3}-2 x^{2}}10x3−2x22x2​, 110x3−1\frac{1}{10 x^{3}-1}10x3−11​, 15x−1\frac{1}{5 x-1}5x−11​, 110x3\frac{1}{10 x^{3}}10x31​.

Problem 302

Identify which of the following expressions equals 1 and which equals -1: x+33+x3−xx−3 \frac{x+3}{3+x} \quad \frac{3-x}{x-3} 3+xx+3​x−33−x​

Problem 303

19. Find the exact value of 3−2.9˙3-2 . \dot{9}3−2.9˙.

Problem 304

1. What is the rational number represented by letter BBB.1. \qquad A) -1.1 B) 0.1 C) -0.1 D) 1.1

Problem 305

Problem 306

Problem 307

Answer ALL questions in the space provided. Show ALL working to receive FULL credit1. Place and label these rational numbers on the number line below: −4.124,−34,312,−118,0.9‾-4.124, \frac{-3}{4}, 3 \frac{1}{2}, \frac{-11}{8}, 0 . \overline{9}−4.124,4−3​,321​,8−11​,0.9 (5 marks)

Problem 308

Calculate the value of 333777333 \sqrt{777}333777​.

Problem 309

What is the expression for Tran's gas cost per mile if he pays \84todrive84 to drive 84todrivem$ miles?

Problem 310

Simplify the expression (a+1a−1+a−1a+1)÷(a+1a−1−a−1a+1)\left(\frac{a+1}{a-1}+\frac{a-1}{a+1}\right) \div\left(\frac{a+1}{a-1}-\frac{a-1}{a+1}\right)(a−1a+1​+a+1a−1​)÷(a−1a+1​−a+1a−1​).

Problem 311

Solve for xxx in the equation xx+3=54\frac{x}{x+3}=\frac{5}{4}x+3x​=45​.

Problem 312

Solve the equation: 4x−4=−4x+3+12(x−4)(x+3)\frac{4}{x-4}=\frac{-4}{x+3}+\frac{12}{(x-4)(x+3)}x−44​=x+3−4​+(x−4)(x+3)12​

Problem 313

Solve for xxx in the equation 3x+3=35x+3110\frac{3}{x}+3=\frac{3}{5 x}+\frac{31}{10}x3​+3=5x3​+1031​.

Problem 314

Solve the inequality: x+3x−1>2\frac{x+3}{x-1}>2x−1x+3​>2

Problem 315

Problem 316

\text{Write the recurring decimal } 99.9\overline{9} \text{ as a fraction in its lowest terms.}

Problem 317

Solve for xxx in the equation gc=x+y−xgc=\frac{x+y}{-x}gc=−xx+y​.

Problem 318

Problem 319

Simplify the expressions and find excluded values: 7. 3x−9x2−6x+9\frac{3 x-9}{x^{2}-6 x+9}x2−6x+93x−9​, 8. 4x−8x2−4x+4\frac{4 x-8}{x^{2}-4 x+4}x2−4x+44x−8​.

Problem 320

Simplify the expression: 6x+93x−15⋅x−54x+6\frac{6 x+9}{3 x-15} \cdot \frac{x-5}{4 x+6}3x−156x+9​⋅4x+6x−5​.

Problem 321

Simplify the expression: x3−1x−3\frac{\frac{x}{3}-1}{x-3}x−33x​−1​.

Problem 322

Add the fractions 8x−2\frac{8}{x-2}x−28​ and 2x−3\frac{2}{x-3}x−32​.

Problem 323

A boat takes 2122 \frac{1}{2}221​ hours and a plane takes 12\frac{1}{2}21​ hour. How much faster is the plane than the boat?

Problem 324

Identify the mistake in 1a+1b=1a+b\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b}a1​+b1​=a+b1​ and correct the right side of the equation.

Problem 325

Find the volume of ethanol that has the same mass as 0.950 L of mercury, given their densities: 13.546 g/mL13.546 \mathrm{~g/mL}13.546 g/mL for mercury and 0.789 g/mL0.789 \mathrm{~g/mL}0.789 g/mL for ethanol.

Problem 326

Convert 0.6 to a simplified fraction. What is the lowest terms fraction for 0.60.60.6?

Problem 327

Multiply and simplify: (−57)(−1415)= \left(-\frac{5}{7}\right)\left(-\frac{14}{15}\right) = (−75​)(−1514​)= (Reduce to lowest terms.)

Problem 328

Multiply and simplify: 211⋅27\frac{2}{11} \cdot \frac{2}{7}112​⋅72​. Provide your answer as an integer or simplified fraction.

Problem 329

Solve for xxx: x+3x+3=1\frac{\sqrt{x+3}}{x+3}=1x+3x+3​​=1. Options: no solution, x=−2x=-2x=−2, x=−3x=-3x=−3, x=−3x=-3x=−3 and x=−2x=-2x=−2.

Problem 330

Find the function (s+t)(x)(s+t)(x)(s+t)(x) for s(x)=x−2x2−25s(x)=\frac{x-2}{x^{2}-25}s(x)=x2−25x−2​ and t(x)=x−52−xt(x)=\frac{x-5}{2-x}t(x)=2−xx−5​. Write the domain in interval notation.

Problem 331

Find (s⋅t)(x)(s \cdot t)(x)(s⋅t)(x) for s(x)=x−3x2−36s(x) = \frac{x-3}{x^{2}-36}s(x)=x2−36x−3​ and t(x)=x−63−xt(x) = \frac{x-6}{3-x}t(x)=3−xx−6​. State the domain in interval notation.

Problem 332

Find (s+t)(x)(s+t)(x)(s+t)(x) for s(x)=x−3x2−36s(x)=\frac{x-3}{x^2-36}s(x)=x2−36x−3​ and t(x)=x−63−xt(x)=\frac{x-6}{3-x}t(x)=3−xx−6​; state the domain in interval notation.

Problem 333

Find (f∘g)(x)(f \circ g)(x)(f∘g)(x) and its domain in interval notation for f(x)=xx−1f(x)=\frac{x}{x-1}f(x)=x−1x​ and g(x)=13x2−36g(x)=\frac{13}{x^{2}-36}g(x)=x2−3613​.

Problem 334

Evaluate (f∘g)(x)(f \circ g)(x)(f∘g)(x) and find its domain in interval notation, where f(x)=xx−1f(x)=\frac{x}{x-1}f(x)=x−1x​ and g(x)=13x2−36g(x)=\frac{13}{x^{2}-36}g(x)=x2−3613​.

Problem 335

Find (f∘g)(x)(f \circ g)(x)(f∘g)(x) for f(x)=xx+9f(x)=\frac{x}{x+9}f(x)=x+9x​ and g(x)=8x2−1g(x)=\frac{8}{x^{2}-1}g(x)=x2−18​; state the domain in interval notation.

Problem 336

Find the domain of (s⋅t)(x)=−1x+6(s \cdot t)(x)=\frac{-1}{x+6}(s⋅t)(x)=x+6−1​, where s(x)=x−3x2−36s(x)=\frac{x-3}{x^{2}-36}s(x)=x2−36x−3​ and t(x)=x−63−xt(x)=\frac{x-6}{3-x}t(x)=3−xx−6​.

Problem 337

Find the domain, range, inverse domain, and inverse range of the function f(x)=54x+1f(x)=\frac{5}{4 x+1}f(x)=4x+15​.

Problem 338

Find the inverse of the one-to-one function f(x)=25x+3f(x)=\frac{2}{5 x+3}f(x)=5x+32​.

Problem 339

Find the inverse of the one-to-one function f(x)=6x+73x+6f(x)=\frac{6 x+7}{3 x+6}f(x)=3x+66x+7​.

Problem 340

Express the number as a ratio of integers. 2.5338‾2.53 \overline{38}2.5338

Problem 341

If 15 boys finish work in 60 days, how many boys are needed for the same work in 20 days?

Problem 342

Simplify $\frac{2 x^{2}}{10 x^{3}-2 x^{2}}$ , $\frac{1}{10 x^{3}-1}$ , $\frac{1}{5 x-1}$ , $\frac{1}{10 x^{3}}$ .

Identify which of the following expressions equals 1 and which equals -1:
$\frac{x+3}{3+x} \quad \frac{3-x}{x-3}$

19. Find the exact value of $3-2 . \dot{9}$ .

1. What is the rational number represented by letter $B$ .
1. $\qquad$ A) -1.1 B) 0.1 C) -0.1 D) 1.1

Answer ALL questions in the space provided. Show ALL working to receive FULL credit
1. Place and label these rational numbers on the number line below: $-4.124, \frac{-3}{4}, 3 \frac{1}{2}, \frac{-11}{8}, 0 . \overline{9}$ (5 marks)

Calculate the value of $333 \sqrt{777}$ .

What is the expression for Tran's gas cost per mile if he pays \ $84 to drive$ m$ miles?

Simplify the expression $\left(\frac{a+1}{a-1}+\frac{a-1}{a+1}\right) \div\left(\frac{a+1}{a-1}-\frac{a-1}{a+1}\right)$ .

Solve for $x$ in the equation $\frac{x}{x+3}=\frac{5}{4}$ .

Solve the equation: $\frac{4}{x-4}=\frac{-4}{x+3}+\frac{12}{(x-4)(x+3)}$

Solve for $x$ in the equation $\frac{3}{x}+3=\frac{3}{5 x}+\frac{31}{10}$ .

Solve the inequality: $\frac{x+3}{x-1}>2$

Solve for $x$ in the equation $gc=\frac{x+y}{-x}$ .

Simplify the expressions and find excluded values: 7. $\frac{3 x-9}{x^{2}-6 x+9}$ , 8. $\frac{4 x-8}{x^{2}-4 x+4}$ .

Simplify the expression: $\frac{6 x+9}{3 x-15} \cdot \frac{x-5}{4 x+6}$ .

Simplify the expression: $\frac{\frac{x}{3}-1}{x-3}$ .

Add the fractions $\frac{8}{x-2}$ and $\frac{2}{x-3}$ .

A boat takes $2 \frac{1}{2}$ hours and a plane takes $\frac{1}{2}$ hour. How much faster is the plane than the boat?

Identify the mistake in $\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b}$ and correct the right side of the equation.

Find the volume of ethanol that has the same mass as 0.950 L of mercury, given their densities: $13.546 \mathrm{~g/mL}$ for mercury and $0.789 \mathrm{~g/mL}$ for ethanol.

Convert 0.6 to a simplified fraction. What is the lowest terms fraction for $0.6$ ?

Multiply and simplify: $\left(-\frac{5}{7}\right)\left(-\frac{14}{15}\right) =$ (Reduce to lowest terms.)

Multiply and simplify: $\frac{2}{11} \cdot \frac{2}{7}$ . Provide your answer as an integer or simplified fraction.

Solve for $x$ : $\frac{\sqrt{x+3}}{x+3}=1$ . Options: no solution, $x=-2$ , $x=-3$ , $x=-3$ and $x=-2$ .

Find the function $(s+t)(x)$ for $s(x)=\frac{x-2}{x^{2}-25}$ and $t(x)=\frac{x-5}{2-x}$ . Write the domain in interval notation.

Find $(s \cdot t)(x)$ for $s(x) = \frac{x-3}{x^{2}-36}$ and $t(x) = \frac{x-6}{3-x}$ . State the domain in interval notation.

Find $(s+t)(x)$ for $s(x)=\frac{x-3}{x^2-36}$ and $t(x)=\frac{x-6}{3-x}$ ; state the domain in interval notation.

Find $(f \circ g)(x)$ and its domain in interval notation for $f(x)=\frac{x}{x-1}$ and $g(x)=\frac{13}{x^{2}-36}$ .

Evaluate $(f \circ g)(x)$ and find its domain in interval notation, where $f(x)=\frac{x}{x-1}$ and $g(x)=\frac{13}{x^{2}-36}$ .

Find $(f \circ g)(x)$ for $f(x)=\frac{x}{x+9}$ and $g(x)=\frac{8}{x^{2}-1}$ ; state the domain in interval notation.

Find the domain of $(s \cdot t)(x)=\frac{-1}{x+6}$ , where $s(x)=\frac{x-3}{x^{2}-36}$ and $t(x)=\frac{x-6}{3-x}$ .

Find the domain, range, inverse domain, and inverse range of the function $f(x)=\frac{5}{4 x+1}$ .

Find the inverse of the one-to-one function $f(x)=\frac{2}{5 x+3}$ .

Find the inverse of the one-to-one function $f(x)=\frac{6 x+7}{3 x+6}$ .

Express the number as a ratio of integers. $2.53 \overline{38}$

Solve for $x$ : $\frac{x+1}{x}+\frac{3}{4}=\frac{-3}{x}$ . What is $x$ ?

Solve $\frac{7}{x}=\frac{2}{3 x}-10$ for $x$ . What is $x$ ?

Solve for $x$ in the equation $\frac{1}{x+4}=\frac{4}{x-4}$ . What is $x$ ?

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+3}$ , $h \neq 0$ . Simplify your answer.

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{2}{x}$ , where $h \neq 0$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{9}{x^{2}}$ , simplifying where $h \neq 0$ .

Solve the inequality $\frac{3 x-5}{-3}>5$ and provide the solution set in interval and graph forms.

Identify which numbers in the set $E=\{\sqrt{14}, \pi, \sqrt{14}+1, \pi+\frac{1}{14}\}$ are (a) Natural, (b) Integers, (c) Rational, (d) Irrational, (e) Real.

Calculate the expression $4 x y^{-1}$ for $x=9$ and $y=-2$ .

Simplify the expression: $\frac{(-5)^{3} x^{7}(y z)^{3}}{(4)^{2} x^{5} y^{7} z^{2}}$

Identify values to exclude from the domain of $\frac{x}{x^{2}-4}$ . Options: $x=0$ , $x=-2$ , $x=-1$ , $x=2$ , None.

Identify values to exclude from the domain of $\frac{x^{2}}{x^{2}+16}$ . Options: $x=5$ , $x=4$ , $x=0$ , $x=-4$ . Select all that apply.

Find values to exclude from the domain of $\frac{x^{2}+7 x-11}{x^{3}-4 x}$ . Options: $x=2$ , $x=11$ , $x=-2$ , $x=0$ . Select all that apply.

Find the domain of $x$ for the expression $\frac{4}{x-19}$ .

Find the domain of $x$ in the expression $\frac{x}{x+79}$ .

Solve the compound inequality: $0.5 < \frac{y+2}{2} < 5.5$ and express the solution in interval notation.

Solve the compound inequality $\frac{22}{8} < \frac{y+7}{4} < \frac{29}{8}$ and express your answer in interval notation. Use decimals.

Solve the inequality: $\frac{3 x-7}{-3}>7$ . Provide the solution set in interval notation and graph form.

Graph the solution set for the compound inequality: $\frac{22}{8}<\frac{y+7}{4}<\frac{29}{8}$ .

Find the value of $\frac{7 \div(q)^{2} \cdot 2}{2 p} \cdot \frac{-p+6 q-r}{-q}$ for $p=4$ , $q=\frac{1}{2}$ , $r=2$ .

Solve the compound inequality $\frac{16}{8}<\frac{y+3}{4}<\frac{25}{8}$ and express the solution in interval notation.

Solve the compound inequality $\frac{16}{8}<\frac{y+3}{4}<\frac{25}{8}$ and express the solution in interval notation.

Solve the compound inequality $\frac{12}{8}<\frac{y+8}{4}<\frac{20}{8}$ and give the answer in interval notation.

Determine the group of sets for the number $-\frac{7}{10}$ . Options: A. real, rational B. real, irrational C. real, rational, natural D. rational, natural, integers E. irrational, natural.

Identify the rational numbers from this list: A. $-\frac{8}{9}$ , B. $\sqrt{2}$ , C. $0 . \overline{25}$ , D. $\pi$ , E. $\frac{2}{1}$ , F. None.

Find the y-intercept of $f(x)=\frac{x^{2}-4}{x+2}$ .

Determine the oblique asymptote for the function $f(x)=\frac{x^{2}-4}{x+2}$ .

Determine the vertical asymptote of $f(x)=\frac{x^{2}-4}{x+2}$ .

Determine the horizontal asymptote for the function $f(x)=\frac{x^{2}-4}{x+2}$ .

Find the holes in the rational function $f(x)=\frac{x^{2}-4}{x+2}$ . If none, state 'none'.

Simplify: $\frac{x^{2}-3 x-4}{x^{2}+x} \div \frac{x-4}{x^{2}}$