What is the value of \dfrac{x + 6}{x - 1} when x = 3?
Hint
Work out the entire numerator and the entire denominator first, then divide.
Step-by-step solution
Substitute x = 3: \dfrac{(3) + 6}{(3) - 1}.
Numerator: 3 + 6 = 9.
Denominator: 3 - 1 = 2.
Divide: \dfrac{9}{2} = 4.5.
Answer: \dfrac{9}{2} (4.5)
Check:4.5 \times 2 = 9, which is the numerator ✓.
Tip: The fraction bar works like parentheses: (x + 6) \div (x - 1). Never divide just part of the top by part of the bottom. Substituting into parentheses first, \dfrac{(3) + 6}{(3) - 1}, keeps every number in the right place.
Question 2 of 36
No calculatorFill in
Evaluate \dfrac{2x^2 - 5}{x + 4} when x = -2.
Enter your answer as a decimal or fraction.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Put -2 in parentheses. Remember that (-2)^2 is positive.
Step-by-step solution
Substitute with parentheses: \dfrac{2(-2)^2 - 5}{(-2) + 4}.
Exponent first: (-2)^2 = 4.
Numerator: 2(4) - 5 = 8 - 5 = 3.
Denominator: -2 + 4 = 2.
Divide: \dfrac{3}{2} = 1.5.
Answer: 1.5 (or \dfrac{3}{2})
Check:1.5 \times 2 = 3 ✓. If you had used -2^2 = -4, the numerator would be -13 and the answer -6.5 — a sign that the parentheses were missing.
Tip: Squaring a negative number always gives a positive result. Write (-2)^2, never -2^2.
Question 3 of 36
No calculator
For which value of x is the expression \dfrac{x + 2}{x^2 - 9} undefined?
Hint
A fraction is undefined when its denominator is 0.
Step-by-step solution
Set the denominator equal to zero: x^2 - 9 = 0.
Factor: (x - 3)(x + 3) = 0.
So x = 3 or x = -3.
Of the choices, only 3 appears.
Answer: x = 3
Check: At x = 3: denominator = 9 - 9 = 0, so \dfrac{5}{0} is undefined ✓. At x = -2: \dfrac{0}{-5} = 0, which is defined.
Tip: Ignore the numerator when looking for undefined values. A zero numerator only makes the whole fraction equal 0.
Question 4 of 36
No calculator
Replace x with 7 in the expression \dfrac{3x - 4}{x + 5} and simplify. What number do you get?
Hint
Find the value of the whole top and the whole bottom first. Only then divide.
Step-by-step solution
Substitute x = 7 using parentheses: \dfrac{3(7) - 4}{(7) + 5}.
Numerator: 3(7) = 21, and 21 - 4 = 17.
Denominator: 7 + 5 = 12.
Divide: \dfrac{17}{12}. This fraction does not reduce, because 17 is prime and does not divide 12.
Answer: \dfrac{17}{12} (about 1.42)
Check:\dfrac{17}{12} \times 12 = 17, which matches the numerator ✓. A value a little more than 1 makes sense, because the top (17) is a little more than the bottom (12).
Tip: Treat the fraction bar as two sets of parentheses: (3x - 4) \div (x + 5). Finish each part completely before you divide.
Question 5 of 36
No calculatorFill in
If x = -6, what does the expression \dfrac{x^2 + 3x}{x - 2} equal?
Write your answer as a decimal or a fraction.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Put -6 in parentheses everywhere you see x. What sign does (-6)^2 have?
Step-by-step solution
Substitute with parentheses: \dfrac{(-6)^2 + 3(-6)}{(-6) - 2}.
Exponent first: (-6)^2 = 36 (a negative times a negative is positive).
Multiply: 3(-6) = -18.
Numerator: 36 + (-18) = 18.
Denominator: -6 - 2 = -8.
Divide: \dfrac{18}{-8} = -\dfrac{9}{4} = -2.25.
Answer: -2.25 (or -\dfrac{9}{4})
Check:-2.25 \times (-8) = 18, the numerator ✓. Positive top, negative bottom, so a negative answer is right.
Tip: If you type -6^2 the calculator gives -36. Always use parentheses: (-6)^2 = 36.
Question 6 of 36
No calculator
Given a = 4 and b = -2, simplify \dfrac{2a + b}{a - b} to a single number.
Hint
Replace b with (-2), including the parentheses. Subtracting a negative is the same as adding.
Step-by-step solution
Substitute: \dfrac{2(4) + (-2)}{(4) - (-2)}.
Numerator: 2(4) = 8, and 8 + (-2) = 6.
Denominator: 4 - (-2) = 4 + 2 = 6.
Divide: \dfrac{6}{6} = 1.
Answer: 1
Check: The top and bottom are the same number, 6, so the fraction must equal 1 ✓. If you get a different denominator, look again at 4 - (-2): the two minus signs make a plus.
Tip: With two variables, write the substitution with parentheses for both before doing any arithmetic. It keeps you from mixing up the values and from losing a negative sign.
Question 7 of 36
A gym charges a one-time $120 sign-up fee plus $35 per month. The average cost per month, in dollars, for a membership that lasts m months is
\frac{120 + 35m}{m}
What is the average cost per month for a membership that lasts 8 months?
Hint
First find the total cost for 8 months (the numerator). Then share it over the 8 months.
Step-by-step solution
Substitute m = 8: \dfrac{120 + 35(8)}{8}.
Multiply: 35 \times 8 = 280.
Numerator (total cost): 120 + 280 = 400 dollars.
Divide by the number of months: \dfrac{400}{8} = 50.
Answer: $50 per month
Check: Another way: \dfrac{120}{8} + 35 = 15 + 35 = 50 ✓. The sign-up fee adds $15 to each of the 8 months.
Tip: In an average-cost formula the numerator is the total cost and the denominator is the number of units. The average is always more than the per-unit rate ($35) because the fixed fee gets spread out.
Question 8 of 36
A print shop charges a $75 setup fee plus $6 per T-shirt. The average cost per shirt, in dollars, for an order of n shirts is
\frac{75 + 6n}{n}
What is the average cost per shirt for an order of 25 shirts?
Hint
Find the whole numerator first (total cost), then divide by the number of shirts.
Step-by-step solution
Substitute n = 25: \dfrac{75 + 6(25)}{25}.
Multiply: 6 \cdot 25 = 150.
Numerator (total cost): 75 + 150 = 225.
Divide by 25: \dfrac{225}{25} = 9.
Answer: $9 per shirt
Check:25 shirts at $9 each is 25 \times 9 = \$225, which equals the $75 fee plus 25 \times \$6 = \$150 ✓.
Tip: The average cost must be more than the $6 per-shirt price (the fee is spread over the shirts) and much less than the total. That quickly rules out $3 and $225.
Question 9 of 36
Fill in
One pump can fill a tank in 4 hours. A second pump can fill the same tank in 12 hours. When both pumps run together, the time in hours to fill the tank is
t = \frac{ab}{a + b}
where a and b are the times for each pump alone. How many hours will the two pumps take together?
hours
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Substitute a = 4 and b = 12. Should the answer be more or less than 4 hours?
Step-by-step solution
Substitute a = 4, b = 12: t = \dfrac{4 \cdot 12}{4 + 12}.
Numerator: 4 \cdot 12 = 48.
Denominator: 4 + 12 = 16.
Divide: t = \dfrac{48}{16} = 3.
Answer: 3 hours
Check: In 3 hours the first pump fills \tfrac{3}{4} of the tank and the second fills \tfrac{3}{12} = \tfrac{1}{4}. Together: \tfrac{3}{4} + \tfrac{1}{4} = 1 full tank ✓.
Tip: Working together is always faster than the faster worker alone, so the answer must be under 4 hours. Averaging the times (\tfrac{4 + 12}{2} = 8) is a classic wrong answer.
Question 10 of 36
Drop-down
Consider the expression \dfrac{x^2 - 16}{x^2 + 4x}.
When x = 4, the value of the expression is .
When x = -4, the expression is .
Choose an answer from each menu.
Hint
Check the numerator and the denominator separately at each value. Which one being zero makes a fraction undefined?
Step-by-step solution
At x = 4: numerator = 16 - 16 = 0; denominator = 16 + 16 = 32.
\dfrac{0}{0} is undefined, because the denominator is zero.
Note: the simplified form \dfrac{x - 4}{x} gives 2 at x = -4, but the original expression still has a zero denominator there, so it is undefined.
Answer: 0; undefined
Check: Factor the denominator: x^2 + 4x = x(x + 4), which is zero at x = 0 and x = -4 ✓. x = 4 is not an excluded value.
Tip: Zero on top is fine (the value is 0). Zero on the bottom is never fine — and canceling a factor does not change that.
Question 11 of 36
Marcus drives to his sister's house at 40 miles per hour and drives home along the same route at 60 miles per hour. His average speed for the whole round trip, in miles per hour, is
\frac{2r_1 r_2}{r_1 + r_2}
where r_1 and r_2 are the two speeds. What is his average speed?
Hint
Substitute both speeds, and compute the whole numerator and the whole denominator before dividing.
Check: Try a 120-mile trip each way: going takes 120 \div 40 = 3 h, returning takes 120 \div 60 = 2 h. Total 240 miles in 5 h = 48 mph ✓.
Tip: The average speed is less than the simple average (50) because Marcus spends more time at the slower speed.
Question 12 of 36
Fill in
Evaluate \dfrac{a^2 - b}{a - b} when a = -3 and b = 5.
Enter your answer as a decimal or fraction.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Put each value in its own parentheses. Watch the sign of a^2 and the sign of a - b.
Step-by-step solution
Substitute: \dfrac{(-3)^2 - (5)}{(-3) - (5)}.
Exponent: (-3)^2 = 9.
Numerator: 9 - 5 = 4.
Denominator: -3 - 5 = -8.
Divide: \dfrac{4}{-8} = -0.5.
Answer: -0.5 (or -\dfrac{1}{2})
Check:-0.5 \times (-8) = 4, the numerator ✓.
Tip: A positive divided by a negative is negative. If your answer is positive, recheck the denominator -3 - 5. Common slips here are writing a^2 = -9 (which gives \dfrac{-14}{-8} = 1.75) or computing -3 - 5 as 2.
Question 13 of 36
Select all
Select all values of x for which \dfrac{x - 5}{x^2 - x - 12} is undefined.
Select all that apply.
Hint
Set the denominator equal to zero and factor it.
Step-by-step solution
Undefined means the denominator is zero: x^2 - x - 12 = 0.
Factor: two numbers that multiply to -12 and add to -1 are -4 and 3, so (x - 4)(x + 3) = 0.
So x - 4 = 0 or x + 3 = 0, giving x = 4 or x = -3.
Tip: The numbers inside the factors have the opposite sign of the solutions: (x - 4) gives x = 4.
Question 14 of 36
No calculatorSelect all
The expression \dfrac{x + 1}{x^2 - 5x} has no value for certain numbers. Choose every number x that makes the expression undefined.
Select all that apply.
Hint
An expression is undefined when its denominator is zero. Try factoring out the common factor in x^2 - 5x.
Step-by-step solution
A fraction is undefined only when its denominator equals 0.
Set the denominator to zero: x^2 - 5x = 0.
Factor out the common factor x: x(x - 5) = 0.
A product is zero when either factor is zero: x = 0 or x - 5 = 0, so x = 5.
The numerator, x + 1, does not matter here. At x = -1 the value is simply \dfrac{0}{6} = 0.
Answer: 0 and 5
Check:x = 0: 0^2 - 5(0) = 0 ✓. x = 5: 25 - 25 = 0 ✓. x = -5: 25 + 25 = 50 \ne 0, so -5 is fine.
Tip: Do not divide both sides of x^2 - 5x = 0 by x — that throws away the solution x = 0. Factor instead.
Question 15 of 36
No calculatorDrop-down
Consider the expression \dfrac{2x - 8}{x^2 - 1}.
When x = 4, the expression .
When x = 1, the expression .
When x = 0, the expression .
Choose an answer from each menu.
Hint
At each value, find the numerator and the denominator separately. Which one being zero makes the expression undefined?
Step-by-step solution
x = 4: numerator 2(4) - 8 = 0; denominator 4^2 - 1 = 15. So the value is \dfrac{0}{15} = 0.
x = 1: numerator 2(1) - 8 = -6; denominator 1^2 - 1 = 0. Dividing by zero is impossible, so the expression is undefined.
x = 0: numerator 2(0) - 8 = -8; denominator 0^2 - 1 = -1. So the value is \dfrac{-8}{-1} = 8.
Answer: equals 0; is undefined; equals 8
Check: The denominator x^2 - 1 = (x - 1)(x + 1) is zero only at x = 1 and x = -1 ✓. So x = 4 and x = 0 give real numbers, and x = 1 does not.
Tip: Zero on top gives a value of 0. Zero on the bottom means undefined. A 0 under the fraction bar is never a value.
Question 16 of 36
In a science lab, two resistors are connected side by side (in parallel). Their combined resistance, in ohms, is
R = \frac{R_1 R_2}{R_1 + R_2}
where R_1 and R_2 are the resistances of the two resistors. What is the combined resistance of a 6-ohm resistor and a 3-ohm resistor connected in parallel?
Hint
Multiply the two resistances for the top. Add them for the bottom.
Step-by-step solution
Substitute R_1 = 6 and R_2 = 3: R = \dfrac{6 \cdot 3}{6 + 3}.
Numerator: 6 \times 3 = 18.
Denominator: 6 + 3 = 9.
Divide: R = \dfrac{18}{9} = 2.
Answer: 2 ohms
Check: In parallel, the combined resistance is always less than the smaller resistor (3 ohms). 2 < 3 ✓. Answers like 9 or 4.5 are larger than 3, so they cannot be right.
Tip: "Product over sum" formulas (\dfrac{ab}{a + b}) show up for resistors, for two pumps or workers, and for lenses. Follow the formula exactly; do not just add or average the inputs.
Question 17 of 36
Fill in
The concentration of a medicine in a patient's blood, in milligrams per liter, t hours after a dose is
C = \frac{50t}{t^2 + 4}
A nurse started this table.
Hours after dose, t
1
2
4
Concentration, C (mg/L)
10
?
10
What is the missing concentration at t = 2 hours?
mg/L
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Substitute t = 2. Square 2 before you add 4.
Step-by-step solution
Substitute t = 2: C = \dfrac{50(2)}{(2)^2 + 4}.
Numerator: 50 \times 2 = 100.
Denominator: 2^2 = 4, and 4 + 4 = 8.
Divide: C = \dfrac{100}{8} = 12.5.
Answer: 12.5 mg/L
Check: Test a known column. t = 1: \dfrac{50}{1 + 4} = 10 ✓. t = 4: \dfrac{200}{16 + 4} = 10 ✓. The level rises and then falls, so a value above 10 at t = 2 makes sense.
Tip: When a table gives you some values already, use one of them to test that you are reading the formula correctly before you fill in the gap.
Question 18 of 36
No calculator
Luis evaluated \dfrac{5 - x^2}{2x + 1} for x = -3. His work is shown.
Step 1:\dfrac{5 - (-3)^2}{2(-3) + 1}
Step 2:\dfrac{5 + 9}{-6 + 1}
Step 3:\dfrac{14}{-5}
Step 4:-2.8
In which step did Luis make his first error?
Hint
Work out (-3)^2 yourself. Then decide what 5 - (\text{that number}) is.
Step-by-step solution
Step 1 is correct: every x is replaced by (-3) in parentheses.
Step 2: (-3)^2 = 9, so the numerator is 5 - 9, not5 + 9. Luis treated the square of -3 as -9 and then subtracted it. This is the first error.
Steps 3 and 4 follow correctly from Step 2, so they are not new errors.
Answer: Step 2 (the correct value is 0.8)
Check:0.8 \times (-5) = -4, the correct numerator ✓.
Tip: Squaring removes the negative sign: (-3)^2 = +9. The minus in front of x^2 belongs to the expression, so 5 - x^2 becomes 5 - 9.
Question 19 of 36
No calculator
Which statement about the expression \dfrac{x + 2}{x - 1} is true?
Hint
Test each statement by substituting its value of x. Watch for a zero on top versus a zero on the bottom.
Step-by-step solution
Test A (x = 1): denominator 1 - 1 = 0, so the expression is undefined, not 0. False.
Test B (x = -2): numerator -2 + 2 = 0, denominator -3. Value \dfrac{0}{-3} = 0, so it is not undefined. False.
Test C (x = 4): \dfrac{4 + 2}{4 - 1} = \dfrac{6}{3} = 2. True.
Test D (x = 0): \dfrac{0 + 2}{0 - 1} = \dfrac{2}{-1} = -2, not 2. False.
Answer: C — When x = 4, the value is 2.
Check:2 \times (4 - 1) = 6 = 4 + 2 ✓.
Tip: For "which statement is true" questions, check every choice quickly. Two of the wrong choices here swap the ideas of "zero" and "undefined" — a favorite GED trap.
Question 20 of 36
No calculator
Kayla made a table of values for the expression \dfrac{60}{x + 2}.
x
-1
1
4
10
\dfrac{60}{x + 2}
60
20
15
5
Which column of the table contains an error?
Hint
Compute x + 2 for each column first, then divide 60 by that number.
Step-by-step solution
x = -1: x + 2 = 1, and \dfrac{60}{1} = 60 ✓.
x = 1: x + 2 = 3, and \dfrac{60}{3} = 20 ✓.
x = 4: x + 2 = 6, and \dfrac{60}{6} = 10. The table shows 15, which is \dfrac{60}{4} — Kayla forgot to add 2. ✗
x = 10: x + 2 = 12, and \dfrac{60}{12} = 5 ✓.
Answer: the column for x = 4 (the value should be 10)
Tip: The whole denominator, x + 2, divides 60. A common slip is dividing by x alone. Also notice that as x grows, the values should shrink steadily: 60, 20, 10, 5.
Question 21 of 36
No calculatorFill in
Evaluate \dfrac{2x^3 + 4}{x^2 - 6} when x = -2.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
A negative number cubed stays negative, but a negative number squared is positive.
Step-by-step solution
Substitute with parentheses: \dfrac{2(-2)^3 + 4}{(-2)^2 - 6}.
Cube: (-2)^3 = (-2)(-2)(-2) = -8.
Numerator: 2(-8) + 4 = -16 + 4 = -12.
Square: (-2)^2 = 4.
Denominator: 4 - 6 = -2.
Divide: \dfrac{-12}{-2} = 6.
Answer: 6
Check:6 \times (-2) = -12, the numerator ✓. Both parts are negative, so the result is positive.
Tip: Odd powers keep the sign ((-2)^3 = -8); even powers make it positive ((-2)^2 = 4). On the calculator, use the (-) key inside parentheses before pressing ^.
Question 22 of 36
A company's average cost, in dollars, to make n phone cases is
\frac{2400 + 8n}{n}
How many phone cases must the company make so that the average cost is $20 per case?
Hint
Set the expression equal to 20 and solve for n, or test each answer choice in the formula.
Step-by-step solution
Set the average cost equal to 20: \dfrac{2400 + 8n}{n} = 20.
What is the correct value, and what did Dana do wrong?
Hint
Can you divide only the 8 by 4 when the numerator is a sum? Try substituting x = 2 into the original expression.
Step-by-step solution
Substitute directly into the original: \dfrac{(2) + 8}{4(2)}.
Numerator: 2 + 8 = 10.
Denominator: 4 \times 2 = 8.
Value: \dfrac{10}{8} = \dfrac{5}{4} = 1.25.
Dana's error: she divided the term 8 by 4 but left the term x alone. You may only cancel a factor that multiplies the entire numerator and the entire denominator. The 4 is a factor of 4x, but not of x + 8.
Answer: 1.25 — Dana canceled the 4 from only one term of the numerator.
Check:1.25 \times 8 = 10 ✓. Dana's answer, 2, would need a numerator of 16.
Tip: If you are not sure a shortcut is legal, just substitute into the original expression. It is never wrong.
Question 25 of 36
Two shops print team banners. For an order of n banners, the average cost per banner, in dollars, is
Shop A: \dfrac{90 + 4n}{n}
Shop B: \dfrac{40 + 6n}{n}
A league needs 30 banners. Which shop has the lower average cost per banner, and by about how much?
Hint
Evaluate both expressions at n = 30, then subtract the smaller from the larger.
Step-by-step solution
Shop A at n = 30: \dfrac{90 + 4(30)}{30} = \dfrac{90 + 120}{30} = \dfrac{210}{30} = 7 dollars.
Shop B at n = 30: \dfrac{40 + 6(30)}{30} = \dfrac{40 + 180}{30} = \dfrac{220}{30} \approx 7.33 dollars.
Shop A is lower. Difference: 7.33 - 7 = 0.33 dollars per banner.
Answer: Shop A, by about $0.33 per banner
Check: Total costs are $210 (A) and $220 (B). The $10 difference spread over 30 banners is \dfrac{10}{30} \approx 0.33 ✓.
Tip: When a question asks about the average cost, compare averages, not totals or setup fees. Shop B has the smaller setup fee, but its higher per-banner charge wins out for a big order.
Question 26 of 36
Fill in
For what value of x does \dfrac{5x + 3}{x - 1} equal 6?
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Set the expression equal to 6 and multiply both sides by x - 1. Distribute the 6 to both terms.
Step-by-step solution
Write the equation: \dfrac{5x + 3}{x - 1} = 6.
Multiply both sides by (x - 1): 5x + 3 = 6(x - 1).
Distribute: 5x + 3 = 6x - 6.
Subtract 5x from both sides: 3 = x - 6.
Add 6 to both sides: x = 9.
Make sure x = 9 does not make the denominator zero: 9 - 1 = 8 \ne 0 ✓.
Tip: "Work backwards" questions turn into equations. Clear the fraction by multiplying by the denominator, and remember to distribute to every term inside the parentheses (6 \cdot xand6 \cdot 1).
Question 27 of 36
No calculator
Let x = \dfrac{1}{2}. Find the exact value of \dfrac{x^2 - 4x}{x^2 - 16}.
Hint
Factor the top and the bottom first. A common factor will cancel, leaving a much simpler expression.
Step-by-step solution
Factor the numerator: x^2 - 4x = x(x - 4).
Factor the denominator (difference of squares): x^2 - 16 = (x - 4)(x + 4).
Cancel the common factor (x - 4) (allowed because x = \tfrac{1}{2} does not make it zero): \dfrac{x}{x + 4}.
Substitute x = \tfrac{1}{2}: \dfrac{\tfrac{1}{2}}{\tfrac{1}{2} + 4} = \dfrac{0.5}{4.5}.
Simplify: \dfrac{0.5}{4.5} = \dfrac{1}{9}.
Answer: \dfrac{1}{9}
Check: Use the original: numerator \tfrac{1}{4} - 2 = -\tfrac{7}{4}; denominator \tfrac{1}{4} - 16 = -\tfrac{63}{4}. Ratio \dfrac{7}{63} = \dfrac{1}{9} ✓.
Tip: When the value of x is messy, look for factors to cancel first. Cancel factors (things multiplied), never separate terms.
Question 28 of 36
No calculator
Which expression is undefined when x = -3?
Hint
Substitute -3 into each denominator only. Which denominator becomes zero?
Step-by-step solution
An expression is undefined when its denominator is zero. Test each denominator at x = -3.
A: x - 3 = -3 - 3 = -6. Not zero. (The numerator is zero, so the value is 0.)
Check: Only the hall fee is shared. Per guest it is \dfrac{1200}{100} = 12 versus \dfrac{1200}{150} = 8; the food cost ($28) is the same either way. 12 - 8 = 4 ✓.
Tip: In "how much less/more" questions, compute both values first and subtract last. Read carefully whether the question wants a total, an average, or a difference.
Question 30 of 36
Using a = -1.5 and b = 0.5, compute \dfrac{a^2 - 2b}{3a + b}.
Hint
Square -1.5 with parentheses. Then find the numerator and denominator separately.
Check:-0.3125 \times (-4) = 1.25 ✓. Positive numerator, negative denominator, so the value is negative.
Tip: On the calculator, type the whole thing with parentheses: ((-)1.5)² − 2(0.5) for the top and 3((-)1.5) + 0.5 for the bottom, or use the n/d template so the fraction bar groups each part.
Question 31 of 36
No calculatorDrag and drop
Complete the table of values for \dfrac{x + 6}{x - 2}. Drag a value into each empty box. Two values are not used.
x
-6
2
4
\dfrac{x + 6}{x - 2}
Tap a tile, then tap a box to place it.
Hint
For each x, check the denominator first. Is it zero?
Step-by-step solution
x = -6: numerator -6 + 6 = 0; denominator -6 - 2 = -8. Value \dfrac{0}{-8} = 0.
x = 2: numerator 2 + 6 = 8; denominator 2 - 2 = 0. Division by zero, so the expression is undefined.
x = 4: numerator 4 + 6 = 10; denominator 4 - 2 = 2. Value \dfrac{10}{2} = 5.
Not used: \tfrac{1}{5} (the value at x = 4 turned upside down) and 8 (the numerator at x = 2 with the zero denominator ignored).
Answer: 0; undefined; 5
Check:5 \times 2 = 10 ✓. And x - 2 = 0 only at x = 2, so that is the only undefined column ✓.
Tip: A table of values for a rational expression usually has one "trap" column where the denominator is zero. Spot it before you calculate.
Question 32 of 36
No calculatorSelect all
Select all true statements about the expression \dfrac{x^2 - 1}{x + 3}.
Select all that apply.
Hint
Test each statement separately. For each value, compute the numerator and the denominator.
Check: Rewrite the expression by multiplying top and bottom by x: \dfrac{x + 1}{x - 1}. At x = \tfrac{1}{3}: \dfrac{\tfrac{4}{3}}{-\tfrac{2}{3}} = -2 ✓.
Tip: In a fraction inside a fraction, work out each small fraction first. "1 divided by one-third" is 3, because three thirds make one whole.
Question 34 of 36
Fill in
When two people work together, the time t (in hours) they need to finish a job is
t = \frac{ab}{a + b}
where a and b are the times each person needs alone. Ana can paint a fence alone in 6 hours. Working together, Ana and Ben paint the fence in 4 hours.
How many hours would Ben need to paint the fence alone?
hours
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Substitute a = 6 and t = 4, leaving b unknown. Then multiply both sides by the denominator.
Step-by-step solution
Substitute what you know: 4 = \dfrac{6b}{6 + b}.
Multiply both sides by (6 + b): 4(6 + b) = 6b.
Distribute: 24 + 4b = 6b.
Subtract 4b: 24 = 2b.
Divide by 2: b = 12.
Answer: 12 hours
Check:\dfrac{6 \cdot 12}{6 + 12} = \dfrac{72}{18} = 4 hours ✓. Also, Ben must be slower than the pair (more than 4 hours), and 12 is.
Tip: To clear a fraction in an equation, multiply both sides by the entire denominator, in parentheses. Then it is an ordinary linear equation.
Question 35 of 36
No calculatorFill in
Compute \dfrac{x^2 - x - 6}{x^2 - 9} for x = -2.5.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Factor both the numerator and the denominator. They share a factor.
Step-by-step solution
Factor the numerator: find two numbers that multiply to -6 and add to -1: -3 and 2. So x^2 - x - 6 = (x - 3)(x + 2).
Factor the denominator (difference of squares): x^2 - 9 = (x - 3)(x + 3).
Cancel (x - 3) — allowed because x = -2.5 does not make it zero: \dfrac{x + 2}{x + 3}.
Tip: Factoring first turns a decimal mess into one easy division. The GED often chooses values like -2.5 precisely so that the simplified form gives a clean answer.
Question 36 of 36
No calculator
Mei wants the value of x that makes \dfrac{2x + 6}{x - 3} equal to 4. Her work is shown.
Step 1:2x + 6 = 4(x - 3)
Step 2:2x + 6 = 4x - 3
Step 3:9 = 2x
Step 4:x = 4.5
Which statement is true?
Hint
Check each step on its own. Pay special attention to how the 4 is distributed.
Step-by-step solution
Step 1 is correct: multiply both sides by (x - 3).
Step 2 is wrong: 4(x - 3) = 4x - 12. Mei multiplied only the x by 4.
Correct work from Step 2: 2x + 6 = 4x - 12, so 18 = 2x, and x = 9.
Steps 3 and 4 follow from Mei's wrong Step 2, so the first error is in Step 2.
Answer: Mei's first error is in Step 2; the correct value is x = 9.
