Tip: Moving the smallerx term (2x) keeps the coefficient positive (4x), which means fewer sign errors. A common mistake here is adding 2x to the left side (8x), which gives the wrong value x = 3.
Question 3 of 36
No calculator
What is the solution of 4(x - 3) = 20?
Hint
Either distribute the 4 to both terms in the parentheses, or divide both sides by 4 first.
Step-by-step solution
Method 1 — distribute first.
Multiply 4 by each term inside: 4 \cdot x - 4 \cdot 3 = 4x - 12, so 4x - 12 = 20.
Add 12 to both sides: 4x = 32.
Divide by 4: x = 8.
Method 2 — divide first. Both sides divided by 4: x - 3 = 5, so x = 8.
The answer is x = 8.
Check:4(8 - 3) = 4(5) = 20 ✓
Tip: When the whole left side is a number times parentheses, dividing first is often faster and avoids distribution errors.
Question 4 of 36
No calculator
If -4x + 9 = -15, then x is equal to which number?
Hint
First undo the +9. Then think carefully about what happens when you divide a negative by a negative.
Step-by-step solution
The left side was built by multiplying x by -4 and then adding 9. Undo those steps in reverse order.
Subtract 9 from both sides: -4x + 9 - 9 = -15 - 9, so -4x = -24.
Divide both sides by -4: x = \dfrac{-24}{-4}.
A negative divided by a negative is positive: x = 6.
The answer is x = 6.
Check:-4(6) + 9 = -24 + 9 = -15 ✓
Tip: When you divide by a negative coefficient, decide the sign first (same signs → positive, different signs → negative), then divide the numbers.
Question 5 of 36
No calculatorFill in
A number x is divided by 5, and then 3 is subtracted. The result is 4:
\frac{x}{5} - 3 = 4
Enter the number x.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Undo the subtraction first. Then ask: what undoes dividing by 5?
Step-by-step solution
Here x was divided by 5 and then 3 was subtracted. Undo in reverse order.
Add 3 to both sides: \dfrac{x}{5} - 3 + 3 = 4 + 3, so \dfrac{x}{5} = 7.
Multiply both sides by 5 (the opposite of dividing by 5): x = 7 \times 5 = 35.
The answer is x = 35.
Check:\dfrac{35}{5} - 3 = 7 - 3 = 4 ✓
Tip: Dividing by a number is undone by multiplying by that same number. A common mistake is dividing 7 by 5 and getting 1.4; that would mean x was multiplied by 5, not divided by 5.
Question 6 of 36
No calculator
Solve for x.
9x + 4 = 5x - 12
Hint
Subtract 5x from both sides so that x appears on only one side.
Step-by-step solution
Subtract 5x from both sides: 9x - 5x + 4 = -12, so 4x + 4 = -12.
Tip: Moving the smaller x term (5x) keeps the coefficient positive. Then be extra careful with the constants: -12 - 4 goes further below zero, to -16, not up to -8.
Question 7 of 36
Solve for x: \dfrac{2}{3}x - 5 = 7
Hint
First get the x term alone. Then remember that dividing by \tfrac{2}{3} is the same as multiplying by its reciprocal.
Step-by-step solution
Add 5 to both sides: \dfrac{2}{3}x = 12.
Multiply both sides by the reciprocal \dfrac{3}{2}: x = 12 \cdot \dfrac{3}{2} = \dfrac{36}{2} = 18.
The answer is x = 18.
Check:\dfrac{2}{3}(18) - 5 = 12 - 5 = 7 ✓
Tip: To undo multiplying by a fraction, multiply by its reciprocal (flip it). Multiplying by \tfrac{2}{3} again is the most common slip — it gives 8, which is too small.
Question 8 of 36
Fill in
Solve for x.
0.6x + 1.5 = 0.2x + 4.7
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Multiply every term by 10 to clear the decimals, or work with the decimals directly. Collect the x terms on one side.
Step-by-step solution
Clear the decimals. Multiply every term on both sides by 10:
Tip: Multiplying by 10 must be done to every term. A common slip is writing 6x + 1.5 (forgetting to multiply the constant), which leads to a wrong answer.
Question 9 of 36
Drop-down
Select the choice that makes the statement true.
The equation 3(2x + 1) = 6x + 3 has .
Choose an answer from each menu.
Hint
Distribute, then try to collect the x terms on one side. What is left when they cancel?
Step-by-step solution
Distribute the 3: 3 \cdot 2x + 3 \cdot 1 = 6x + 3, so the equation becomes 6x + 3 = 6x + 3.
Subtract 6x from both sides: 3 = 3.
The variable is gone and the statement left over is true. It does not depend on x at all, so every value of x works.
The equation has infinitely many solutions.
Check: try two values. x = 0: 3(1) = 3 and 0 + 3 = 3 ✓. x = 5: 3(11) = 33 and 30 + 3 = 33 ✓.
Tip: When the variables cancel, look at what is left. True (3 = 3) → infinitely many solutions. False (like 3 = 1) → no solution. Never write "x = 0" just because the x's disappeared.
Question 10 of 36
What is the value of x?
5 - 2(x + 4) = 3x + 2
Hint
Multiply -2 by both terms in the parentheses before doing anything else. Do not subtract 5 - 2 first.
Step-by-step solution
Order of operations says multiplication comes before subtraction, so distribute the -2 first.
Distribute: -2 \cdot x = -2x and -2 \cdot 4 = -8. The equation becomes 5 - 2x - 8 = 3x + 2.
Combine like terms on the left: 5 - 8 = -3, so -2x - 3 = 3x + 2.
Tip: Multiplying by the LCD turns a fraction equation into an equation with whole numbers. Remember to multiply the number on the right side too: forgetting it gives 5x = 15 and the wrong answer x = 3.
Question 16 of 36
What value of x makes the equation true?
1.2(x + 5) = 0.4x + 10
Hint
Distribute 1.2 to both terms, then collect the x terms on the left.
Step-by-step solution
Distribute: 1.2 \cdot x + 1.2 \cdot 5 = 1.2x + 6. The equation is 1.2x + 6 = 0.4x + 10.
Subtract 0.4x from both sides: 0.8x + 6 = 10.
Subtract 6 from both sides: 0.8x = 4.
Divide by 0.8: x = \dfrac{4}{0.8} = 5.
The answer is x = 5.
Check: left 1.2(5 + 5) = 1.2(10) = 12; right 0.4(5) + 10 = 2 + 10 = 12 ✓
Tip: If decimals bother you, multiply every term by 10 first: 12(x + 5) = 4x + 100. You get the same answer with whole numbers.
Question 17 of 36
For what value of x are these two fractions equal?
Cross-multiply: each numerator is multiplied by the denominator on the other side. Keep the parentheses.
Step-by-step solution
Cross-multiply: 3(2x - 1) = 5(x + 4).
Distribute on both sides: 6x - 3 = 5x + 20.
Subtract 5x from both sides: x - 3 = 20.
Add 3 to both sides: x = 23.
The answer is x = 23.
Check: left \dfrac{2(23) - 1}{5} = \dfrac{45}{5} = 9; right \dfrac{23 + 4}{3} = \dfrac{27}{3} = 9 ✓
Tip: Write the parentheses when you cross-multiply. The number multiplies the whole numerator, not just its first term.
Question 18 of 36
No calculator
Which equation has no solution?
Hint
Simplify both sides of each equation. Look for a result like 5 = 6, which is never true.
Step-by-step solution
Simplify each equation.
A:4(x - 2) = 4x - 8 becomes 4x - 8 = 4x - 8. Both sides are identical, so everyx works (infinitely many solutions).
B: the right side is 3(x + 1) + 2 = 3x + 3 + 2 = 3x + 5. So 3x + 6 = 3x + 5. Subtract 3x: 6 = 5, which is false for every x. No solution.
C:7x + 1 = 7 + x. Subtract x: 6x + 1 = 7, so 6x = 6 and x = 1. One solution.
D:5x = 0 gives x = 0. One solution (zero is a real answer).
The answer is B: 3x + 6 = 3(x + 1) + 2.
Check: try x = 0 in B: 6 vs. 3(1) + 2 = 5. Try x = 10: 36 vs. 35. The right side is always 1 less, so the sides never match ✓
Tip: Same x coefficient but different constants → no solution. Same coefficient and same constant → infinitely many.
Question 19 of 36
Select all
Select all the equations that are true when x = -4.
Select all that apply.
Hint
Substitute -4 for x in each equation. Put the -4 in parentheses so you keep track of signs.
Step-by-step solution
Substitute x = -4 into each equation and compare both sides.
A:3(-4) + 20 = -12 + 20 = 8 ✓ true.
B:-2((-4) - 1) = -2(-5) = 10 ✓ true.
C:\dfrac{-4}{2} + 6 = -2 + 6 = 4, not 8 ✗ (this equation's solution is x = 4).
D:5 - (-4) = 5 + 4 = 9, not 1 ✗ (this equation's solution is x = 4).
E:0.5(-4) - 3 = -2 - 3 = -5 ✓ true.
The correct choices are A, B, and E.
Check: solve A directly: 3x = 8 - 20 = -12, so x = -4 ✓. Solve E: 0.5x = -2, so x = -4 ✓.
Tip: On "select all" questions, test every choice. Substituting is often faster than solving each equation, and parentheses around the negative value prevent sign mistakes.
Question 20 of 36
No calculator
Devon solved the equation 4x - 3(x - 2) = 10 as shown.
Step
Work
1
4x - 3x - 6 = 10
2
x - 6 = 10
3
x = 16
In which step did Devon make his first error?
Hint
Check each step separately. What is (-3) \times (-2)?
Step-by-step solution
Check each step.
Step 1: The -3 must multiply both terms in (x - 2): (-3)(x) = -3x and (-3)(-2) = +6. The correct line is 4x - 3x + 6 = 10. Devon wrote -6, so Step 1 is wrong.
Steps 2 and 3 do the right operations on Devon's wrong line, so they only carry the error forward.
Correct solution: 4x - 3x + 6 = 10, so x + 6 = 10, so x = 4.
Tip: Simplify each side on its own first (combine like terms), then move terms across the equals sign. Doing both at once is where most mistakes happen.
Question 23 of 36
Fill in
What number can replace x to make 0.35x - 1.4 equal to 0.15x + 2.6?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Collect the decimal x terms on the left and the constants on the right. Multiplying every term by 100 is another option.
Step-by-step solution
Subtract 0.15x from both sides: 0.20x - 1.4 = 2.6.
Add 1.4 to both sides: 0.2x = 4.0.
Divide by 0.2: x = \dfrac{4}{0.2} = 20.
Another way: multiply every term by 100: 35x - 140 = 15x + 260. Then 20x = 400, so x = 20.
Tip: Dividing by a decimal like 0.2 is the same as multiplying by 5. On the calculator, enter 4 \div 0.2 to be sure.
Question 24 of 36
Fill in
Solve for x.
\frac{3}{4}(x - 8) + 2 = \frac{1}{2}x + 1
x =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Multiply every term on both sides by 4 to clear the fractions. Then distribute and collect the x terms.
Step-by-step solution
The denominators are 4 and 2, so the LCD is 4. Multiply every term by 4:
4 \cdot \frac{3}{4}(x - 8) + 4 \cdot 2 = 4 \cdot \frac{1}{2}x + 4 \cdot 1 \;\Rightarrow\; 3(x - 8) + 8 = 2x + 4
Tip: When you clear fractions, the plain numbers (2 and 1) get multiplied by the LCD too. Forgetting them is the most common error on this type of question.
Question 25 of 36
Select all
Select all the equations that have x = -2 as their solution.
Select all that apply.
Hint
Substitute -2 for x in each equation, or solve each one. Use parentheses around -2.
Step-by-step solution
Substitute x = -2 into each equation (or solve it):
Check: By solving E, 4x - 2x = -7 + 3 \Rightarrow 2x = -4 \Rightarrow x = -2 ✓
Tip: On "select all" questions, test every option — there is usually more than one correct choice. Substituting is faster than solving each equation from scratch.
Question 26 of 36
Fill in
The variable is in the denominators of this proportion: \dfrac{3}{x - 2} = \dfrac{5}{x + 4}. What value of x makes it true?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Cross-multiply, keeping each denominator in parentheses.
Step-by-step solution
Cross-multiply: 3(x + 4) = 5(x - 2).
Distribute: 3x + 12 = 5x - 10.
Subtract 3x from both sides: 12 = 2x - 10.
Add 10 to both sides: 22 = 2x.
Divide by 2: x = 11.
Note that x = 11 does not make either denominator zero (11 - 2 = 9 and 11 + 4 = 15), so it is allowed.
The answer is x = 11.
Check:\dfrac{3}{9} = \dfrac{1}{3} and \dfrac{5}{15} = \dfrac{1}{3} ✓
Tip: When the variable is in a denominator, cross-multiplying still gives a linear equation. Always check that your answer does not make a denominator equal zero.
Question 27 of 36
Drop-down
Select the choice that makes each statement true.
Equation 1: 2(4x - 1) + 3 = 8x + 1 has .
Equation 2: 5(x + 2) - x = 4x + 1 has .
Choose an answer from each menu.
Hint
Simplify the left side of each equation completely, then compare it with the right side.
Step-by-step solution
Equation 1.
Distribute: 8x - 2 + 3 = 8x + 1.
Combine constants: 8x + 1 = 8x + 1.
Both sides are identical. Subtracting 8x gives 1 = 1, which is always true → infinitely many solutions.
Equation 2.
Distribute: 5x + 10 - x = 4x + 1.
Combine like terms: 4x + 10 = 4x + 1.
Subtract 4x: 10 = 1, which is never true → no solution.
Answers: Equation 1 has infinitely many solutions; Equation 2 has no solution.
Check: try x = 2 in each. Eq. 1: 2(7) + 3 = 17 and 8(2) + 1 = 17 ✓ (works, like every number). Eq. 2: 5(4) - 2 = 18 and 4(2) + 1 = 9 ✗ (fails, like every number).
Tip: When the x terms cancel, look at what is left: a true statement means every number works; a false statement means no number works.
Question 28 of 36
No calculatorDrag and drop
Kira is solving 2(x - 3) + 4x = 18. Drag an equation into each box to show her steps in order. Two equations are not used.
Step 1 (distribute):
Step 2 (combine like terms):
Step 3 (undo the subtraction):
Step 4 (undo the multiplication):
Tap a tile, then tap a box to place it.
Hint
Do each step yourself on your erasable note board, then find the matching tile.
Step-by-step solution
Distribute the 2 to both terms: 2x - 6 + 4x = 18.
Combine like terms (2x + 4x = 6x): 6x - 6 = 18.
Add 6 to both sides: 6x = 24.
Divide both sides by 6: x = 4.
The unused tiles show common mistakes: 2x - 3 + 4x = 18 multiplies only the x by 2, and 6x = 12 subtracts 6 instead of adding it.
Order: 2x - 6 + 4x = 18, then 6x - 6 = 18, then 6x = 24, then x = 4.
Check:2(4 - 3) + 4(4) = 2 + 16 = 18 ✓
Tip: Each correct step gives an equation that is equivalent to the one before it, so every line has the same solution, x = 4. You can test any line by substituting 4.
Question 29 of 36
Solve for x.
\frac{2}{5}(x + 10) = \frac{1}{3}x + 6
Hint
Multiply every term on both sides by 15, the LCD of 5 and 3. Keep the parentheses on the left.
Step-by-step solution
The LCD of 5 and 3 is 15. Multiply every term by 15:
15 \cdot \dfrac{2}{5}(x + 10) = 15 \cdot \dfrac{1}{3}x + 15 \cdot 6.
Simplify: 6(x + 10) = 5x + 90.
Distribute: 6x + 60 = 5x + 90.
Subtract 5x from both sides: x + 60 = 90.
Subtract 60: x = 30.
The answer is x = 30.
Check: left \dfrac{2}{5}(40) = 16; right \dfrac{1}{3}(30) + 6 = 10 + 6 = 16 ✓
Tip: When clearing fractions, the LCD multiplies every term, including the plain number 6. Then distribute as usual.
Question 30 of 36
No calculator
The letter c stands for a fixed number in the equation 6x + c = 2(3x - 5). Which value of c makes the equation true for every value of x (infinitely many solutions)?
Hint
Simplify the right side. For infinitely many solutions, the two sides must be exactly the same expression.
Step-by-step solution
Distribute on the right: 2(3x - 5) = 6x - 10. The equation is 6x + c = 6x - 10.
The x terms already match (6x on both sides). Subtract 6x: c = -10.
If c = -10, both sides are 6x - 10, so everyx works: infinitely many solutions.
If c is any other number, you get a false statement such as 3 = -10: no solution.
The answer is c = -10.
Check: with c = -10, try x = 1: left 6 - 10 = -4, right 2(3 - 5) = -4 ✓. Try x = 5: left 20, right 2(10) = 20 ✓.
Tip: Infinitely many solutions: same x coefficient and same constant. No solution: same x coefficient but different constants.
Question 31 of 36
Fill in
In the equation below, a is a constant.
ax - 7 = 2x + 9
The solution of the equation is x = 4. What is the value of a?
a =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
You know x. Put 4 in for x everywhere, and the only unknown left is a.
Step-by-step solution
Because x = 4 is the solution, substituting 4 for x must make the equation true.
Substitute x = 4: a(4) - 7 = 2(4) + 9.
Simplify the right side: 4a - 7 = 8 + 9 = 17.
Add 7 to both sides: 4a = 24.
Divide by 4: a = 6.
The answer is a = 6.
Check: with a = 6 the equation is 6x - 7 = 2x + 9. Subtract 2x: 4x - 7 = 9, so 4x = 16 and x = 4 ✓
Tip: When a problem gives you the solution and asks for a constant, substitute the solution first. That turns it into an ordinary equation in the constant.
Question 32 of 36
Theo solved 0.5(6x + 4) = 2x - 3 as shown.
Step
Work
1
3x + 2 = 2x - 3
2
5x + 2 = -3
3
5x = -5
4
x = -1
Which statement is true?
Hint
Check Step 1 by distributing 0.5 yourself. Then check how Theo moved the 2x term in Step 2.
Step-by-step solution
Check each step.
Step 1:0.5 \cdot 6x = 3x and 0.5 \cdot 4 = 2, so 3x + 2 = 2x - 3 is correct.
Step 2: To remove 2x from the right side, subtract 2x from both sides: 3x - 2x + 2 = -3, which is x + 2 = -3. Theo added2x to the left and got 5x. Step 2 is wrong.
Correct finish: x + 2 = -3, so x = -5.
The true statement is: Step 2 is wrong, and the correct solution is x = -5.
Check:x = -5: left 0.5(6(-5) + 4) = 0.5(-26) = -13; right 2(-5) - 3 = -13 ✓. Theo's answer x = -1: left 0.5(-2) = -1, right -5 ✗.
Tip: A term "moves" across the equals sign by doing the opposite operation on both sides. +2x on the right is removed by subtracting 2x.
Question 33 of 36
No calculator
Which equation is equivalent to the equation below and contains no fractions?
\frac{x}{4} - \frac{2}{3} = \frac{5}{6}
Hint
Find the LCD of 4, 3, and 6, and multiply each of the three terms by it.
Step-by-step solution
The LCD of 4, 3, and 6 is 12.
Multiply every term by 12:
12 \cdot \dfrac{x}{4} = 3x
12 \cdot \dfrac{2}{3} = 8
12 \cdot \dfrac{5}{6} = 10
The new equation is 3x - 8 = 10.
The answer is 3x - 8 = 10.
Check: solve both. New equation: 3x = 18, so x = 6. Original: \dfrac{6}{4} - \dfrac{2}{3} = \dfrac{18}{12} - \dfrac{8}{12} = \dfrac{10}{12} = \dfrac{5}{6} ✓. Same solution, so they are equivalent.
Tip: To multiply a fraction by the LCD, divide the LCD by the denominator, then multiply by the numerator: 12 \div 3 = 4, 4 \times 2 = 8.
Question 34 of 36
In the equation below, k is a constant.
4(2x - 3) = kx + 5
For what value of k does the equation have no solution?
Hint
Distribute first. An equation has no solution when the x terms cancel but the constants do not match.
Step-by-step solution
Distribute the 4: 4(2x - 3) = 8x - 12. The equation is 8x - 12 = kx + 5.
If k is not 8, you could subtract kx and divide by 8 - k, giving exactly one solution: x = \dfrac{17}{8 - k}.
If k = 8, the equation becomes 8x - 12 = 8x + 5. Subtracting 8x from both sides leaves -12 = 5, which is false. No value of x can make it true.
The answer is k = 8.
Check: with k = 8, try x = 0: -12 vs 5; try x = 1: -4 vs 13. The right side is always 17 more than the left, so they are never equal ✓
Tip: No solution = samex-coefficient on both sides but different constants. (If the constants matched too, there would be infinitely many solutions.)
Question 35 of 36
Drop-down
Marisol solved the equation \dfrac{x}{3} + 2 = \dfrac{x}{2} - 1 as shown.
Step
Work
1
Multiply both sides by 6: 2x + 2 = 3x - 1
2
Subtract 2x: 2 = x - 1
3
Add 1: x = 3
Select the choice that makes each statement true.
Marisol's first error is in . The correct solution is x = .
Choose an answer from each menu.
Hint
When you multiply both sides by 6, which terms must be multiplied? Check what happened to the 2 and the -1.
Step-by-step solution
Find the error.
Multiplying both sides by 6 means multiplying every term by 6: 6 \cdot \dfrac{x}{3} = 2x, 6 \cdot 2 = 12, 6 \cdot \dfrac{x}{2} = 3x, 6 \cdot (-1) = -6.
The correct Step 1 is 2x + 12 = 3x - 6. Marisol left the 2 and the -1 unchanged, so Step 1 is wrong. Steps 2 and 3 are done correctly on her wrong line.
Solve correctly.
2x + 12 = 3x - 6.
Subtract 2x: 12 = x - 6.
Add 6: x = 18.
The first error is in Step 1, and the correct solution is x = 18.
