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Questions and Answers

When subtracting fractions with like denominators, which step is crucial after finding a common denominator?

  • Divide the denominators.
  • Subtract the numerators. (correct)
  • Add the denominators.
  • Multiply the numerators.

What is the simplified form of $p^{-1}q^2$?

  • $\frac{p}{q^2}$
  • $\frac{1}{pq^2}$
  • $pq^2$
  • $\frac{q^2}{p}$ (correct)

Given the expression $\frac{7p}{pq^{-2}} - \frac{2}{q}$, what is the first step to simplify it?

  • Eliminate the variable $p$.
  • Factor out common terms.
  • Combine the denominators.
  • Write the expression with only positive exponents. (correct)

After rewriting $\frac{7p}{pq^{-2}} - \frac{2}{q}$ with positive exponents, what is the resulting expression before simplification?

<p>$\frac{7pq^2}{q} - \frac{2}{q}$ (B)</p> Signup and view all the answers

What is the simplified form of $\frac{7pq^2}{q} - \frac{2}{q}$?

<p>$\frac{7pq^2 - 2}{q}$ (A)</p> Signup and view all the answers

A rectangular field has a length of $\frac{6a}{3a+4}$ km and a width of $\frac{8}{3a+4}$ km. What expression represents the perimeter of the field?

<p>$2(\frac{6a}{3a+4} + \frac{8}{3a+4})$ (B)</p> Signup and view all the answers

If the perimeter of a rectangular field is given by $2(\frac{6a}{3a+4} + \frac{8}{3a+4})$, what is the simplified expression for the perimeter?

<p>$\frac{12a + 16}{3a+4}$ (B)</p> Signup and view all the answers

Given the expression $\frac{12a + 16}{3a+4}$, what is the completely simplified form of this expression?

<p>$4$ (D)</p> Signup and view all the answers

The area of a store's sign cannot exceed 900 square decimeters, according to a city bylaw. Which inequality represents the possible areas, $A$, of the sign?

<p>$A \leq 900$ (C)</p> Signup and view all the answers

Given the numbers 176, 134, 208, 170, 149, 153, 136, 200, 168, 150, 157, 141, 211, 176, 145, 155, 128, 199, 182, 148, what is the median of this data set?

<p>153.5 (D)</p> Signup and view all the answers

Rewrite the equation $5x + 3y = 9$ in slope-intercept form.

<p>$y = -\frac{5}{3}x + 3$ (A)</p> Signup and view all the answers

Which equation has a slope of $-\frac{1}{2}$ and a y-intercept of -3?

<p>$5x + 10y + 30 = 0$ (D)</p> Signup and view all the answers

Which values from the set ${-1, 0, 1, 2}$ are solutions to the inequality $x - 1 \geq 4$?

<p>None (C)</p> Signup and view all the answers

The graph shows the temperature is now at least the temperature it was this morning. How can the current temperature be expressed as a sentence given the inequality $x \ge -14$?

<p>The current temperature is greater than or equal to -14 degrees. (D)</p> Signup and view all the answers

The specifications for a housing development include that each lot has to be at least $3\frac{1}{2}$ acres. Which inequality represents this situation, where $A$ is the size of a lot in acres?

<p>$A \geq 3.5$ (D)</p> Signup and view all the answers

What is the first step in determining the equation of a line given two points?

<p>Calculate the slope. (B)</p> Signup and view all the answers

Given the simple interest formula I = prt, how would tripling the principal (p) and halving the time (t) affect the total interest (I) earned?

<p>The interest will be 1.5 times the original interest. (B)</p> Signup and view all the answers

For what value(s) of x is the rational expression $\frac{x - 5}{15x^2 - 75x}$ undefined?

<p>x = 0 and x = 5 (D)</p> Signup and view all the answers

A line has a slope of 3 and passes through the points (-1, -2) and (4, y). What is the value of y?

<p>13 (D)</p> Signup and view all the answers

Which inequality correctly translates the sentence: "The difference of -4 and an unknown number is less than or equal to 0"?

<p>$-4 - x \le 0$ (B)</p> Signup and view all the answers

What is the value of $- \sqrt{x}$ when $x = \frac{25}{16}$?

<p>$\frac{-5}{4}$ (D)</p> Signup and view all the answers

Arminda has a bag business. It costs her $1100 plus $18 per bag to produce them, and she sells the bags for $40 each. How many bags must she sell to gain a profit of at least $2200?

<p>150 bags (D)</p> Signup and view all the answers

Given the dataset of newborns' weights: 5, 8, 6, 5, 7, 8, 10, 7, 8, 6. What are the mean, median, and mode, respectively?

<p>6.8, 7, 8 (C)</p> Signup and view all the answers

What is the slope of the line that passes through the points (-1, 2) and (3, 2)?

<p>0 (C)</p> Signup and view all the answers

Rachel crochets 24 inches of a 72-inch blanket, and Michelle finishes it. If Michelle crochets at 8 inches per day, which equation represents the blanket's length, $y$, after Michelle crochets for $x$ days?

<p>$y = 24 + 8x$ (A)</p> Signup and view all the answers

John is 500 miles from home and traveling towards it at a constant rate of 65 miles per hour. The equation $d = 500 - 65t$ gives his distance $d$ from home after $t$ hours. How far away from home is John after 4 hours?

<p>240 miles (A)</p> Signup and view all the answers

Which of the following expressions is a monomial?

<p>$-5x^2yz^3$ (A)</p> Signup and view all the answers

Determine the degree of the monomial: $7a^3b^2c$

<p>6 (A)</p> Signup and view all the answers

Which expression represents the total length Michelle needs to crochet to complete the blanket?

<p>48 inches (C)</p> Signup and view all the answers

Consider John's trip home. If he travels for 5 hours, what does the value of $d$ represent in the equation $d = 500 - 65t$?

<p>John's remaining distance from home. (A)</p> Signup and view all the answers

A monomial has a degree of 5 and contains the variables $x$ and $y$. Which of the following could represent this monomial?

<p>$4x^4y$ (D)</p> Signup and view all the answers

A company approximates its national advertising spending with the polynomial $59x^2 - 262x + 3888$ and local advertising with the polynomial $-33x^3 + 611x^2 - 1433x + 28060$, where $x$ represents years since the campaign began. What polynomial represents the total advertising spending?

<p>$-33x^3 + 670x^2 - 1695x + 31948$ (D)</p> Signup and view all the answers

Given the points (4, 5) and (6, 4) lie on a line, what is the equation of this line in slope-intercept form?

<p>$y = -\frac{1}{2}x + 7$ (A)</p> Signup and view all the answers

Which expression is not a polynomial?

<p>$\frac{1}{x}$ (A)</p> Signup and view all the answers

What is the degree of the polynomial $-a^2b^2c^3 + 5x^5$?

<p>7 (B)</p> Signup and view all the answers

What polynomial, when subtracted from $3x^2 + 7x - 6$, results in $-6$?

<p>$3x^2 + 7x$ (D)</p> Signup and view all the answers

What is the equation of the line in point-slope form that passes through the point (3, 1) and has a slope of -1?

<p>$y - 1 = -1(x - 3)$ (C)</p> Signup and view all the answers

If a line passes through the point (6, -3) and has slope of -2, which of the following points also lies on this line?

<p>(4, 1) (C)</p> Signup and view all the answers

Is the statement $4 = 4x^0$ true or false, and why?

<p>True, because any number to the power of zero is one. (B)</p> Signup and view all the answers

Given the distances of planets from the sun (in millions of miles): 36, 67, 93, 142, 484, 887, 1765, and 2791, which of the following values would be considered an outlier when constructing a box-and-whisker plot?

<p>1765 (D)</p> Signup and view all the answers

A product's price increased from $2175.00 to $2392.50. What is the approximate percentage increase in the price?

<p>10% (D)</p> Signup and view all the answers

For the quiz scores: 86, 92, 88, 100, 86, 94, 92, 78, 90, 96, 94, 84, which measure of central tendency would be most appropriate to represent the data, considering potential outliers and distribution?

<p>Median (A)</p> Signup and view all the answers

A skateboard factory has 467 skateboards in stock and produces 115 skateboards per hour. Assuming no shipments are made, which linear equation in slope-intercept form represents the number of skateboards in inventory after $x$ hours?

<p>$y = 115x + 467$ (A)</p> Signup and view all the answers

A tire's tread depth decreases as mileage increases. Given the data points (10000 miles, 20 mm), (20000 miles, 16 mm), (30000 miles, 12 mm), and (40000 miles, 8 mm), what is the average rate of wear for this tire brand, in mm per 10,000 miles?

<p>4 mm (C)</p> Signup and view all the answers

Consider the statements: (1) If a vehicle is a truck, then the vehicle is automatic. (2) All trucks are motor vehicles. Which of the following is true?

<p>Statement 1 is false; statement 2 is true. (D)</p> Signup and view all the answers

What is the key difference between $\sqrt{-1}$ and $\sqrt{1}$?

<p>$\sqrt{-1}$ is an imaginary number, while $\sqrt{1}$ is a real number. (C)</p> Signup and view all the answers

Why cannot the expression $\frac{2g}{2g + 6}$ be simplified to $\frac{1}{6}$?

<p>Because you can only cancel common factors, not common terms, in a fraction. (D)</p> Signup and view all the answers

Flashcards

Subtracting Rational Expressions

To subtract rational expressions, they must have a common denominator. Then, subtract the numerators and keep the denominator.

Inequality

A mathematical statement that compares two expressions using inequality symbols.

Graph of an Inequality

A visual representation of all solutions to an inequality on a number line.

Positive Exponents

A method to eliminate negative exponents by moving the term to the opposite side of the fraction bar.

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X-intercept

The point where a line crosses the x-axis.

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Greatest Common Factor (GCF)

The greatest factor that divides two or more numbers. Factoring it out simplifies the expression.

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Perimeter

The sum of the lengths of all sides of a two-dimensional shape.

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Y-intercept

The point where a line crosses the y-axis.

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Proportion

A comparison between two ratios.

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Rectangle

A quadrilateral with two pairs of parallel sides.

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Slope

A measure of the steepness and direction of a line.

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Perimeter of a Rectangle Formula

P = 2(l + w), where 'l' is the length and 'w' is the width.

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Slope-intercept form

A linear equation in the form y = mx + b, where m is slope and b is the y-intercept.

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Substitution

Replacing variables in an expression with their given values.

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Median

The middle value in a sorted list of numbers.

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Simplification

To reduce to a simpler form by canceling common factors.

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Zero Slope

The slope is zero when the y-coordinates are the same.

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What is a Monomial?

A monomial is a product of numbers and/or variables with whole number exponents.

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Degree of a Monomial

The sum of the exponents of the variables in the monomial. Constants have a degree of 0.

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Like Terms

Terms that have the same variable factors raised to the same powers.

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Adding Polynomials

To add polynomials, combine like terms (add coefficients).

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Subtracting Polynomials

To subtract polynomials, distribute the negative sign and then combine like terms.

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Term (in an Expression)

A part of an expression that is added to or subtracted from the other parts.

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Simple Interest

The amount of money the borrower pays based on the principal for a given period of time.

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Undefined Rational Expression

Values of a variable where the denominator of a rational expression equals zero.

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Mode

The value that appears most frequently in a data set.

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Radical Index

Part of a radical expression, indicating the root to be taken.

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Point-Slope Form

An equation of a line written as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

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Polynomial

A mathematical expression involving variables and coefficients, that only uses addition, subtraction, and non-negative integer exponents.

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Degree of a Polynomial

The highest exponent of a variable in a polynomial expression.

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Zero Exponent Rule

Any non-zero number raised to the power of 0 is equal to 1.

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Box-and-Whisker Plot

A visual representation of data using quartiles, median, and extremes.

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Ordering Data

Listing data points in order from least to greatest is a common error to ommit or include.

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Percent Increase/Decrease

The percent change from an original value to a new value.

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Measure of Central Tendency

A single value that best represents a set of data (mean, median, or mode).

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Average Rate of Wear

The amount of change over a period of time.

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Counterexample

A statement that is not true.

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√-1 vs √1

√-1 is imaginary, √1 is real.

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Study Notes

  • Rational expressions include variables in the denominator.
  • All values that would make the denominator equal to zero are excluded.

Identifying Excluded Values

  • To identify excluded values set the Denominator = 0, and solve for variables

Simplifying Rational Expressions

  • Factor out the greatest common factor (GCF).
  • Divide out common factors.
  • If there is more than one term in the numerator or denominator, begin by factoring

Simplifying Expressions with Integer Exponents

  • Write the expression with only positive exponents.
  • The expressions must have like denominators in order to subtract the numerators.
  • Factor the numerator, and the GCF is 2. Simplify after factorisation

Application: Fencing a Field

  • Use the formula for finding the perimeter of a rectangle: P = 2 (l + w)
  • Substitute the length and width.
  • If like denominators exist, add the numerators.

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