Mathematics and ELAR Review Practice Flashcards

Flashcards created to review key concepts in mathematics and English Language Arts & Reading (ELAR) from lecture materials.

What is the simplified value of the expression $15 - 3 \times (4 + 2)^2 \div 9$ using the Order of Operations (PEMDAS)?

The correct value is $3$. Following PEMDAS: 1. Parentheses: $(4 + 2) = 6$. 2. Exponents: $6^2 = 36$. 3. Multiplication/Division (left to right): $3 \times 36 = 108$, then $108 \div 9 = 12$. 4. Subtraction: $15 - 12 = 3$.

What are the formulas for the Product Rule and Quotient Rule when simplifying exponential expressions with the same base $x$?

Product Rule: $x^a \cdot x^b = x^{a+b}$. Quotient Rule: $\frac{x^a}{x^b} = x^{a-b}$.

How do you calculate the sum of $\frac{1}{4} + \frac{2}{3}$?

The final result is $\frac{11}{12}$. First, find the Least Common Denominator (LCD), which is $12$. Convert $\frac{1}{4}$ to $\frac{3}{12}$ by multiplying numerator and denominator by $3$. Convert $\frac{2}{3}$ to $\frac{8}{12}$ by multiplying numerator and denominator by $4$. Then add the new numerators: $\frac{3}{12} + \frac{8}{12} = \frac{11}{12}$.

If a recipe calls for $3\,\text{cups}$ of water for every $2\,\text{cups}$ of rice, how many cups of water do you need for $8\,\text{cups}$ of rice?

You need $12\,\text{cups}$ of water. Set up the proportion $\frac{3}{2} = \frac{x}{8}$, cross-multiply to get $3 \times 8 = 2 \times x$ ($24 = 2x$), and divide by $2$ to solve for $x = 12$.

What is the Percent Proportion formula used to find what percentage a part is of a whole?

The Percent Proportion is $\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}$.

How do you convert the decimal $0.375$ into its lowest-term fraction?

Read $0.375$ as "three hundred seventy-five thousandths", write it as $\frac{375}{1000}$, and divide both numerator and denominator by their Greatest Common Divisor ($125$) to get $\frac{3}{8}$.

What is the solution to the two-step equation $3x - 7 = 14$?

The solution is $x = 7$. First, isolate the variable term by adding $7$ to both sides ($3x = 21$). Second, divide both sides by $3$ ($x = 7$).

What crucial rule must be remembered when multiplying or dividing both sides of an inequality by a negative number?

You MUST reverse (flip) the inequality sign. For example, dividing both sides of $-2x < 6$ by $-2$ yields $x > -3$.

What is the value of $y$ in the system of equations $y = 2x + 1$ and $3x + y = 11$?

The value of $y$ is $5$. Using substitution, substitute $2x + 1$ for $y$ in the second equation: $3x + (2x + 1) = 11 \implies 5x + 1 = 11 \implies 5x = 10 \implies x = 2$. Then substitute $x = 2$ back into $y = 2(2) + 1$ to get $y = 5$.

What is the formula for calculating the Area ($A$) of a trapezoid with height ($h$) and parallel bases ($b_1$ and $b_2$)?

The formula is $A = \frac{1}{2}(b_1 + b_2)h$.

What is the slope ($m$) of the line passing through the points $(2, 3)$ and $(6, 11)$?

The slope is $m = 2$. Apply the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ using $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (6, 11)$: $m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$.

In a right triangle with legs of length $a = 5$ and $b = 12$, what is the length of the hypotenuse ($c$)?

The length of the hypotenuse is $c = 13$. Apply the Pythagorean Theorem $a^2 + b^2 = c^2$: $5^2 + 12^2 = c^2 \implies 25 + 144 = c^2 \implies 169 = c^2 \implies c = \sqrt{169} = 13$.

What does the FOIL acronym stand for when multiplying two binomials?

FOIL stands for First (multiply the First terms), Outer (multiply the Outer terms), Inner (multiply the Inner terms), and Last (multiply the Last terms).

What is the Quadratic Formula used to solve for $x$ in any equation of the form $ax^2 + bx + c = 0$?

The Quadratic Formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

When is a semicolon (;) correctly used to join two clauses in a sentence?

A semicolon is used to join two complete sentences (independent clauses) that are closely related in meaning without using a coordinating conjunction.

What component is required alongside a comma when joining two independent clauses in a compound sentence?

A coordinating conjunction from the FANBOYS acronym (For, And, Nor, But, Or, Yet, So) must accompany the comma. Joining two independent clauses with only a comma creates an incorrect comma splice.

How do you ensure correct subject-verb agreement in sentences containing intervening prepositional phrases, such as "The box of pencils is on the table"?

Mentally cross out all prepositional phrases coming between the subject and the verb (e.g., "of pencils") to isolate the true subject ("box", singular) and match it with a singular verb ("is").

What are the definitions of the three primary rhetorical appeals: Ethos, Pathos, and Logos?

Ethos is the appeal to credibility, character, or authority; Pathos is the appeal to emotion; and Logos is the appeal to logic, facts, and structured reasoning.

What is the difference between Active Voice and Passive Voice in academic writing?

In Active Voice, the subject performs the action of the verb (e.g., "The scientist conducted the experiment"). In Passive Voice, the subject receives the action (e.g., "The experiment was conducted by the scientist"). Active voice is preferred for clarity and conciseness.

What is a Thesis Statement, where is it located in an essay, and what three key qualities must it possess?

A Thesis Statement is a single sentence located at the end of the introductory paragraph that expresses the main argument or claim of the entire essay. A strong thesis must be arguable, specific, and controllable.