Middle School Math Quiz: Understand Solving Equations And Inequalities

20 questions · exam conditions

0:00

Understand Solving Equations And InequalitiesQuestion 1 of 20

Sarah claims that $x = 3$ is a solution to the inequality $5x - 7 ≤ 2x + 2$ because when she substitutes, she gets $8 ≤ 8$ , which is true. However, when she solves the inequality algebraically, she gets $x ≤ 3$ . Which statement best explains this situation?

Sarah made an error in her algebraic solution since substitution shows

x = 3

works perfectly

Sarah's substitution confirms that

x = 3

is indeed a solution, and it's the boundary value

Sarah made an error in substitution since

5(3) - 7 = 7

, not

8

as she calculated

Sarah's algebraic work is wrong because inequalities cannot have boundary values like

x = 3

← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Understand Solving Equations And Inequalities

Practice Understand Solving Equations And Inequalities in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Understand Solving Equations And Inequalities, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Sarah made an error in her algebraic solution since substitution shows $x = 3$ works perfectly
Sarah's substitution confirms that $x = 3$ is indeed a solution, and it's the boundary value (correct answer)
Sarah made an error in substitution since $5(3) - 7 = 7$ , not $8$ as she calculated
Sarah's algebraic work is wrong because inequalities cannot have boundary values like $x = 3$

Explanation: Sarah's work is completely correct. When

x = 3

5(3) - 7 = 15 - 7 = 8

and

2(3) + 2 = 8

, so

8 ≤ 8

is true. Solving algebraically:

5x - 7 ≤ 2x + 2

, so

3x ≤ 9

, thus

x ≤ 3

. The value

x = 3

is the boundary value where equality holds. Choice A is wrong because both methods agree. Choice C is wrong because

5(3) - 7 = 8

. Choice D is wrong because inequalities can include boundary values.

Question 2

A game gives bonus points based on $x$ . From the set $\{3,4,5,6\}$ , which values make the equation $x+2=8$ true?

$\{5\}$
$\{4,5\}$
$\{6\}$ (correct answer)
No values in the set make the equation true.

Question 3

Marcus is checking which values from the set { $-2, -1, 0, 1, 2$ } make the inequality $3x + 4 > x - 2$ true. He substitutes each value and gets these results: For $x = -2$ : $-2 > -6$ (true), For $x = -1$ : $1 > -3$ (true), For $x = 0$ : $4 > -2$ (true), For $x = 1$ : $7 > -1$ (true), For $x = 2$ : $10 > 0$ (true). What error did Marcus make?

He substituted the values into the wrong side of the inequality in each case
He simplified $3x + 4$ incorrectly when $x$ was negative in the first two cases
He substituted the values into the original inequality instead of solving it first algebraically
He evaluated $x - 2$ incorrectly, writing $-6$ instead of $-4$ when $x = -2$ (correct answer)

Explanation: When

x = -2

, Marcus should get

x - 2 = -2 - 2 = -4

, not

-6

. The correct comparison should be

-2 > -4

(true). Marcus made an arithmetic error. Choice A is wrong because he substituted correctly into both sides. Choice B is wrong because

3(-2) + 4 = -2

and

3(-1) + 4 = 1

are correct. Choice C is wrong because substitution is the correct method for this problem.

Question 4

Two students are checking if $x = -2$ is a solution to $3(x + 4) = 2x + 8$ . Student A gets: $3(-2 + 4) = 3(2) = 6$ and $2(-2) + 8 = 4$ , so $6 = 4$ (false). Student B gets: $3(-2 + 4) = 3(2) = 6$ and $2(-2) + 8 = -4 + 8 = 4$ , so $6 = 4$ (false). What can you conclude?

Both students made the same arithmetic error, but their conclusion that $x = -2$ is not a solution is actually correct
Student A made an error in computing $2(-2) + 8$ , while Student B computed everything correctly and reached the right conclusion
Both students computed correctly and properly concluded that $x = -2$ does not satisfy the given equation (correct answer)
Student B made an error in computing $3(-2 + 4)$ , while Student A computed everything correctly and reached the right conclusion

Explanation: Both students performed the substitution correctly. For the left side:

3(-2 + 4) = 3(2) = 6

. For the right side:

2(-2) + 8 = -4 + 8 = 4

. So the equation becomes

6 = 4

, which is false. Therefore

x = -2

is not a solution. Both students reached the correct conclusion through correct arithmetic. Student A's work shown is incomplete but the final answer is right.

Question 5

Test each value in the set $\{1,2,3,4,5,6\}$ to find all solutions to the inequality $x>3$ .

$\{1,2,3\}$
$\{3,4,5,6\}$
$\{4,5,6\}$ (correct answer)
No values in the set satisfy the inequality.

Explanation: This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x > 3 and the test set {1,2,3,4,5,6}, where you determine which values satisfy it by substituting each one. For example, for x>3 from the set, test each: 4>3 true, 5>3 true, 6>3 true, but 1>3 false, 2>3 false, 3>3 false (since > excludes equal), so solutions are {4,5,6}; inequalities can have multiple solutions, unlike equations which typically have one. For instance, checking fully: 1>3 false, 2>3 false, 3>3 false, 4>3 true, 5>3 true, 6>3 true, confirming {4,5,6}. The correct solution is identified by substitution as {4,5,6}. A common error is including the boundary incorrectly, like adding 3 when > excludes it, or not testing all values and omitting some true ones. To test systematically: (1) list the set {1,2,3,4,5,6}, (2) substitute each into x>3 and check, (3) mark true values like 4,5,6, (4) collect solutions as {4,5,6}; remember the inequality direction, as x>3 excludes 3 but includes greater values.

Question 6

From the set $\{0,1,2,3,4\}$ , which values make the inequality $2x<6$ true? (Check each value by substitution.)

$\{0,1,2,3,4\}$
$\{1,2,3\}$
$\{0,1,2,3\}$
$\{0,1,2\}$ (correct answer)

Explanation: This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality 2x<6 and the test set {0,1,2,3,4}, where you determine which values satisfy it by substitution: pick a value like x=2, replace (22<6 becomes 4<6), evaluate and check if true—yes, so x=2 is a solution; inequalities can have multiple solutions, unlike equations which typically have one. For example, with inequality 2x<6 and set {0,1,2,3,4}, check x=0:0<6 true, x=1:2<6 true, x=2:4<6 true, x=3:6<6 false, x=4:8<6 false, so solutions are {0,1,2}. Correct solution identification comes from substituting, multiplying, and checking the inequality. Common errors include including boundary like x=3 when 6<6 is false, arithmetic errors like 23=5, or not testing all. Testing systematically: (1) list the set {0,1,2,3,4}, (2) test each like x=3:6<6 false, (3) mark true values like x=0,1,2, (4) collect solutions {0,1,2}. Remember < means strictly less, excluding equality.

Question 7

From the set $\{0,1,2,3,4,5\}$ , which values make the inequality $2x<10$ true? (Test each value.)

$\{0,1,2,3,4\}$ (correct answer)
$\{1,2,3,4,5\}$
$\{0,1,2,3,4,5\}$
$\{5\}$

Explanation: Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the inequality 2x<10 with the set {0,1,2,3,4,5}, the solving process involves determining which values satisfy it by substituting each one. Substitution: for x=4, 2(4)=8<10 true; for x=5, 2(5)=10<10 false since 10 is not less than 10. In contrast, an equation like x+5=12 has typically one solution. For this inequality, test: 2(0)=0<10 true, 2(1)=2<10 true, 2(2)=4<10 true, 2(3)=6<10 true, 2(4)=8<10 true, 2(5)=10<10 false, so solutions {0,1,2,3,4}. The correct solution set is identified by substitution, excluding the boundary where it's equal. A common error is including x=5 by mistake, thinking 10<10 is true, or arithmetic errors in multiplication.

Question 8

Which values from the set $\{1,2,3,4,5,6,7\}$ make the inequality $2x\le 10$ true? (Substitute each value to check.)

$\{1,2,3,4\}$
$\{5,6,7\}$
$\{1,2,3,4,5,6,7\}$
$\{1,2,3,4,5\}$ (correct answer)

Explanation: This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality 2x ≤ 10 and the test set {1,2,3,4,5,6,7}, where you determine which values satisfy it by substituting each one. For example, x=5:2(5)=10≤10 true, x=1:2≤10 true, but x=6:12≤10 false; includes boundary for ≤. For instance, 2(1)=2≤10 true, 2(2)=4≤10 true, 2(3)=6≤10 true, 2(4)=8≤10 true, 2(5)=10≤10 true, 2(6)=12≤10 false, 2(7)=14≤10 false, so {1,2,3,4,5}. The correct solution is identified by substitution as {1,2,3,4,5}. A common error is excluding boundary like omitting 5 when 10=10, or including beyond. To test systematically: (1) list {1,2,3,4,5,6,7}, (2) substitute into 2x≤10, (3) mark true up to 5, (4) collect {1,2,3,4,5}.

Question 9

A student says, "Solving $x+4=11$ means finding the value(s) of $x$ that make the equation true." From the set $\{5,6,7,8\}$ , which value(s) make $x+4=11$ true? (Test by substitution.)

$\{6\}$
$\{7\}$ (correct answer)
$\{5\}$
$\{5,6,7,8\}$

Explanation: Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the equation x+4=11 with the set {5,6,7,8}, the solving process involves determining which values satisfy it by substituting each one. Substitution: x=7:7+4=11=11 true, while x=5:5+4=9≠11 false, x=6:6+4=10≠11 false, x=8:8+4=12≠11 false. In contrast, an inequality might have multiple solutions, but this has only one. So the solution is {7}. The correct value is identified by substitution. A common error is arithmetic, like 7+4=12, or claiming multiple values.

Question 10

For which value from the set { $4, 5, 6, 7$ } does the equation $2(x - 3) = x + 1$ become false when you substitute it?

Only $x = 4$ makes the equation false since substitution gives $2 = 5$
Only $x = 7$ makes the equation false since substitution gives $8 = 8$
Values $x = 4, 5, 6$ all make the equation false when properly substituted (correct answer)
Values $x = 5, 6, 7$ all make the equation false when properly substituted

Explanation: Substituting each value: For

x = 4

2(4-3) = 2(1) = 2

and

4 + 1 = 5

, so

2 = 5

(false). For

x = 5

2(5-3) = 4

and

5 + 1 = 6

, so

4 = 6

(false). For

x = 6

2(6-3) = 6

and

6 + 1 = 7

, so

6 = 7

(false). For

x = 7

2(7-3) = 8

and

7 + 1 = 8

, so

8 = 8

(true). Only

x = 7

makes the equation true, so

x = 4, 5, 6

all make it false.

Question 11

From the set { $-1, 0, 1, 2$ }, which value(s) make the inequality $x^2 + 1 ≤ 2$ true when substituted?

All four values from the set satisfy the inequality since squaring always produces small positive results
Only $x = -1$ and $x = 1$ satisfy the inequality when their squares are computed and substituted
Only $x = 0$ satisfies the inequality since it's the only value that makes $x^2 = 0$
Values $x = -1, 0,$ and $1$ all satisfy the inequality when substituted and computed correctly (correct answer)

Explanation: Substituting each value into

x^2 + 1 ≤ 2

: For

x = -1

(-1)^2 + 1 = 1 + 1 = 2

, and

2 ≤ 2

is true. For

x = 0

0^2 + 1 = 1

, and

1 ≤ 2

is true. For

x = 1

1^2 + 1 = 2

, and

2 ≤ 2

is true. For

x = 2

2^2 + 1 = 5

, and

5 ≤ 2

is false. So

x = -1, 0, 1

all work.

Question 12

Consider the equation $2x - 3 = x + 1$ . If you solve this algebraically, you get $x = 4$ . Now test whether each value from the set { $2, 3, 4, 5$ } actually satisfies the original equation by substitution. What do you discover?

Only $x = 4$ satisfies the equation, confirming the algebraic solution, while other values create false statements (correct answer)
Values $x = 3$ and $x = 4$ both satisfy the equation, suggesting the algebraic solution missed one possibility
All values except $x = 2$ satisfy the equation, indicating an error in the algebraic solution process
Values $x = 4$ and $x = 5$ both satisfy the equation, confirming that linear equations can have multiple solutions

Explanation: Testing by substitution: For

x = 2

2(2) - 3 = 1

and

2 + 1 = 3

, so

1 = 3

(false). For

x = 3

2(3) - 3 = 3

and

3 + 1 = 4

, so

3 = 4

(false). For

x = 4

2(4) - 3 = 5

and

4 + 1 = 5

, so

5 = 5

(true). For

x = 5

2(5) - 3 = 7

and

5 + 1 = 6

, so

7 = 6

(false). Only

x = 4

works, confirming the algebraic solution.

Question 13

From the set $\{3,4,5,6\}$ , which values make the inequality $x\ge 5$ true? (Test each value.)

$\{3,4,5\}$
$\{4,5\}$
$\{6\}$
$\{5,6\}$ (correct answer)

Explanation: Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the inequality x≥5 with the set {3,4,5,6}, the solving process involves determining which values satisfy it by substituting each one. Substitution: x=5:5≥5 true (equal counts), x=6:6≥5 true, x=3:3≥5 false, x=4:4≥5 false. In contrast, an equation has typically one solution. So solutions are {5,6}. The correct set is identified by substitution, including the boundary for ≥. A common error is excluding 5, thinking ≥ is like >, or not testing all.

Question 14

A teacher asks students to find which values from { $0, 1, 2, 3$ } satisfy $\frac{x + 4}{2} = x + 1$ . One student claims that $x = 2$ works because $\frac{2 + 4}{2} = 3$ and $2 + 1 = 3$ , so both sides equal $3$ . Is this student correct, and what about the other values?

The student is correct about $x = 2$ , and additionally $x = 0$ also satisfies the equation when checked
The student is correct about $x = 2$ , but it's the only value from the set that satisfies the equation (correct answer)
The student made an error since $\frac{2 + 4}{2} = 4$ , not $3$ , so $x = 2$ doesn't actually work
The student is correct about $x = 2$ , and both $x = 1$ and $x = 3$ also work when substituted properly

Explanation: The student's work for

x = 2

is correct:

\frac{2 + 4}{2} = \frac{6}{2} = 3

and

2 + 1 = 3

, so

3 = 3

is true. Checking other values: For

x = 0

\frac{0 + 4}{2} = 2

and

0 + 1 = 1

, so

2 = 1

(false). For

x = 1

\frac{1 + 4}{2} = 2.5

and

1 + 1 = 2

, so

2.5 = 2

(false). For

x = 3

\frac{3 + 4}{2} = 3.5

and

3 + 1 = 4

, so

3.5 = 4

(false). Only

x = 2

works.

Question 15

Compare the two equations using the test value $x=4$ .

Equation 1: $x+2=6$
Equation 2: $2x=6$

Which statement is true?

$x=4$ is a solution to both equations.
$x=4$ is a solution to Equation 1 only. (correct answer)
$x=4$ is not a solution to either equation.
$x=4$ is a solution to Equation 2 only.

Explanation: Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For testing x=4 in Equation 1: x+2=6 and Equation 2: 2x=6, the process involves substituting into each. Substitution: for Eq1, 4+2=6=6 true; for Eq2, 2(4)=8=6 false. In contrast, if it were an inequality, multiple might work, but here it's equations. So x=4 is a solution to Equation 1 only. The correct statement is identified by separate substitutions. A common error is miscalculating, like 2(4)=6, or confusing the equations.

Question 16

A student is testing whether values from { $1, 2, 3, 4$ } satisfy the compound inequality $2 ≤ x + 1 < 5$ . After substitution, which values from the set make both parts of the compound inequality true?

Only values $1$ and $4$ satisfy both inequality conditions when substituted and evaluated
Only values $2$ and $3$ satisfy both inequality conditions when substituted and evaluated
Values $1, 2,$ and $3$ all satisfy both inequality conditions when substituted and evaluated (correct answer)
Values $2, 3,$ and $4$ all satisfy both inequality conditions when substituted and evaluated

Explanation: For

2 ≤ x + 1 < 5

, substituting each value: For

x = 1

2 ≤ 1 + 1 < 5

becomes

2 ≤ 2 < 5

(true). For

x = 2

2 ≤ 2 + 1 < 5

becomes

2 ≤ 3 < 5

(true). For

x = 3

2 ≤ 3 + 1 < 5

becomes

2 ≤ 4 < 5

(true). For

x = 4

2 ≤ 4 + 1 < 5

becomes

2 ≤ 5 < 5

(false, since

5 < 5

is false). Therefore,

x = 1, 2, 3

all work.

Question 17

A teacher writes $3x=15$ and gives the possible values $\{3,4,5,6\}$ . Which value(s) from the set make the equation true? (Test by substitution.)

$\{4,5\}$
$\{5\}$ (correct answer)
No value in the set makes it true
$\{3\}$

Explanation: This question tests understanding of solving equations as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given equation 3x=15 and the test set {3,4,5,6}, where you determine which values satisfy it by substitution: pick a value like x=5, replace (35=15), evaluate (15=15), and check if true—yes, so x=5 is a solution; for inequalities like x>4 from {2,3,4,5,6}, test each to find {5,6}, noting equations typically have one solution while inequalities can have multiple. For example, with equation 3x=15 and set {3,4,5,6}, check x=3:33=9≠15 false, x=4:34=12≠15 false, x=5:35=15=15 true, x=6:36=18≠15 false, so only x=5 is the solution. Correct solution identification comes from substituting each value and verifying equality after multiplication. Common errors include wrong values like x=3 when 33≠15, arithmetic mistakes like 35=18, or claiming no solution when there is one. Testing systematically: (1) list the set {3,4,5,6}, (2) test each like x=4:34=12≠15 false, (3) mark true values like x=5 true, (4) collect solutions {5}. No solution if none work, but here x=5 does.

Question 18

From the set $\{2,3,4,5\}$ , which values make the inequality $x<4$ true? (Test each value.)

$\{2,3,4\}$
$\{2,3\}$ (correct answer)
$\{4,5\}$
$\{2,3,4,5\}$

Explanation: This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x<4 and the test set {2,3,4,5}, where you determine which values satisfy it by substitution: pick a value like x=3, replace (3<4), evaluate and check if true—yes, so x=3 is a solution; inequalities can have multiple solutions from the set. For example, with inequality x<4 and set {2,3,4,5}, check x=2:2<4 true, x=3:3<4 true, x=4:4<4 false, x=5:5<4 false, so solutions are {2,3}. Correct solution identification comes from testing each and confirming strict inequality. Common errors include including x=4 when < excludes equality, or not testing all values. Testing systematically: (1) list the set {2,3,4,5}, (2) test each like x=4:4<4 false, (3) mark true values like x=2 and x=3, (4) collect solutions {2,3}. Remember < means less than, not less than or equal.

Question 19

From the set $\{2,3,4,5,6\}$ , find the solution set for the inequality $x>4$ . (Check each value to see if the statement is true.)

$\{5,6\}$ (correct answer)
$\{2,3,4\}$
$\{2,3,4,5,6\}$
$\{4,5,6\}$

Explanation: Understanding solving equations and inequalities means finding which values from a specified set make the statement true, using substitution to check each value. For the inequality x>4 with the set {2,3,4,5,6}, the solving process involves determining which values satisfy it by substituting each one. For example, try x=5: 5>4 is true, x=6:6>4 true, but x=4:4>4 false since it's equal, not greater; x=2:2>4 false. In contrast, an equation like x+5=12 typically has one solution, such as x=7. For this inequality x>4 with set {2,3,4,5,6}, check x=2:2>4 false, x=3:3>4 false, x=4:4>4 false, x=5:5>4 true, x=6:6>4 true, so solutions are {5,6}. The correct solution set is identified by substitution, and here it's {5,6}. A common error is including the boundary like 4 when the inequality is strict (>), or not testing all values.

Question 20

A student is checking solutions by substitution. Which values from the set $\{4,5,6,7\}$ make the equation $x+8=14$ true?

$\{4,5\}$
$\{6\}$ (correct answer)
$\{7\}$
No values in the set make the equation true.

Explanation: This question tests understanding of solving equations as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given equation x + 8 = 14 and the test set {4,5,6,7}, where you determine which values satisfy it by substituting each one. For example, try x=6: replace the variable to get 6 + 8 = 14, evaluate to 14 = 14, which is true, so x=6 is a solution; equations like this typically have one solution, unlike inequalities which can have multiple. For instance, checking the set: x=4 gives 4+8=12≠14 false, x=5 gives 5+8=13≠14 false, x=6 gives 6+8=14=14 true, x=7 gives 7+8=15≠14 false, so only x=6 is the solution. The correct solution is identified by substitution as {6}. A common error is an arithmetic mistake, like claiming x=5 works because 5+8=14 incorrectly, or not testing all values and missing the solution. To test systematically: (1) list the set {4,5,6,7}, (2) substitute each into x+8=14 and evaluate, (3) mark true values like x=6, (4) collect solutions as {6}; remember, if no value works, report no solution in the set.