Expressions & Meaning
In 7x - 4, what do we call 7, x, and -4?
7 is the coefficient, x is the variable, and -4 is the constant term.
In 5³, identify the base and exponent. What does the expression mean?
Base: 5. Exponent: 3. It means 5 × 5 × 5, or three factors of 5.
What feature turns 4x + 7 into an equation, and what does that feature mean?
An equals sign connecting it to another expression. It states that the two sides have the same value, for example 4x + 7 = 19.
Which number-line endpoint belongs to x ≤ 4: an open circle or a closed circle? Why?
A closed circle at 4, because 4 is included. Shade to the left.
In y = mx + b, what do m and b represent?
m is the slope, or change in y per unit change in x. b is the y-intercept value, so the line passes through (0, b).
What makes two terms like terms? Use 3x² and 3x to explain.
Like terms have exactly the same variable part, including exponents. 3x² and 3x are not like terms because their exponents differ.
How do the exponent rules differ for x³ × x⁴ and (x³)⁴?
For a product with the same base, add exponents: x⁷. For a power of a power, multiply exponents: x¹².
Why must you subtract 5 from both sides of 2x + 5 = 13 rather than only from the left?
Applying the same subtraction to both sides preserves equality and the solution. Changing only one side changes the equation.
When does an inequality sign reverse while solving? Does subtracting a negative number trigger that rule?
Reverse the sign when multiplying or dividing both sides by a negative number. Subtracting a negative does not reverse the sign.
What is an ordered-pair solution of a two-variable equation? How do you check one?
It is an (x, y) pair that makes the equation true. Substitute both coordinates into the original equation and check that the two sides match.
A student says 3(n + 2) and 3n + 2 mean the same thing. Explain the difference in words and identify the error.
3(n + 2) means add 2 to n, then triple the entire sum. 3n + 2 means triple n, then add 2. The student failed to multiply the 2 inside the parentheses by 3.
A student claims that x⁻³ means -x³. Explain the error and give a numerical counterexample.
A negative exponent means reciprocal: x⁻³ = 1/x³ for x ≠ 0. At x = 2, x⁻³ = 1/8, while -x³ = -8.
Explain why expanding 3(x + 2) to 3x + 6 on one side of an equation does not require distributing anything on the other side.
Expansion replaces an expression with an equivalent expression, so that side keeps the same value. This differs from adding, subtracting, multiplying, or dividing to change a side’s value, which requires a matching operation on both sides.
Explain the difference between x > -2 AND x ≤ 3, and x > -2 OR x ≤ 3. Describe both solution sets.
AND requires both conditions, giving (-2, 3]. OR requires at least one condition. Every real number satisfies at least one here, so the OR solution is all real numbers.
Why is a horizontal line’s slope zero while a vertical line’s slope is undefined? Use rise/run in your explanation.
On a horizontal line, the rise is 0 and the run between distinct points is nonzero, so slope = 0. On a vertical line, the run is 0, so calculating slope would require division by zero.
A student tests x = 0 and concludes that x² + x and 2x are equivalent because both equal 0. Is this enough evidence? Explain and give a counterexample.
No. Equivalent expressions must match for every allowed value. At x = 2, x² + x = 6 but 2x = 4. One matching input does not prove equivalence.
Use the quotient rule to explain why a⁰ = 1, and state the restriction on a.
For a ≠ 0, a³/a³ = 1 because a nonzero quantity divided by itself is 1. The quotient rule also gives a³/a³ = a⁰. Therefore a⁰ = 1. The argument fails at a = 0 because division by zero is undefined.
An equation simplifies to 0 = 0. Another simplifies to 0 = 7. What does each tell you about its solutions, and why?
0 = 0 is true for every real x, so the first has all real numbers as solutions. 0 = 7 is always false, so the second has no solution. The variable terms have canceled in both cases.
A student writes the interval for x ≥ -3 as (-3, ∞]. Identify and explain both endpoint mistakes.
-3 must use a bracket because it is included. Infinity must use a parenthesis because it is not a reachable real endpoint. The correct interval is [-3, ∞).
A table has points (0, 1), (1, 3), and (3, 7). A student says it is not linear because y increases by 2, then 4. Explain the flaw and justify the correct conclusion.
Compare change in y with change in x. The slopes are 2/1 = 2 and 4/2 = 2. The rate is constant, so the points lie on a line. Unequal x-steps can produce unequal y-steps.
In -2(3x - 4) + x, a student reports three terms: -2, 3x - 4, and x. Explain the term-versus-factor error, then identify the terms before and after simplifying.
Before expanding, the two additive terms are -2(3x - 4) and x. Inside the first term, -2 and (3x - 4) are factors. Distributing and combining gives -5x + 8, whose terms are -5x and 8.
A student writes (2x²y)³ = 6x⁵y³. Identify both errors and explain the correct rule for every factor.
The coefficient must be cubed: 2³ = 8, not 2 × 3. A power of a power multiplies exponents: (x²)³ = x⁶, not x⁵. Also y³ remains. The correct result is 8x⁶y³.
A student divides 5x = 2x by x, obtains 5 = 2, and concludes there is no solution. Explain why that step can lose a solution. Solve correctly and check the value.
Dividing by x assumes x ≠ 0, which discards x = 0. Instead subtract 2x from both sides: 3x = 0, so x = 0. Check: 5(0) = 2(0). Division is valid only when the divisor is nonzero.
Without solving by routine algebra, compare |x - 2| < 5 and |x - 2| > 5 using distance. Explain AND versus OR, the boundary points, and whether the boundaries are included.
Both measure distance from 2. Less than 5 gives the inside region: -3 < x < 7, an AND statement. Greater than 5 gives x < -3 OR x > 7, the outside regions. The boundaries -3 and 7 are excluded because both inequalities are strict.
For points (-1, 5) and (3, -3), a student uses (-3 - 5)/(-1 - 3) and gets a positive slope. Explain the error, give the correct slope, and describe the graph’s direction.
The numerator uses second minus first, but the denominator reverses that order. Use (-3 - 5)/(3 - (-1)) = -8/4 = -2. The graph decreases from left to right. Reversing both subtraction orders would also give -2.