Practice Equations and inequalities practice: 6 questions with answers Six equations and inequalities practice questions, from easy to hard, with worked explanations. Pick an answer to check it instantly.
These questions come from our SHSAT, TACHS and SSAT practice apps. For parents
An equation says two things are equal. To solve it, keep it balanced: whatever you do to one side, do to the other.
The balance method
Undo addition and subtraction first. For 3x + 5 = 20, subtract 5 from both sides: 3x = 15.Then undo multiplication and division. Divide both sides by 3: x = 5.Check by substituting. 3(5) + 5 = 20, so x = 5 is right.
When you multiply or divide both sides by a negative number, flip the inequality sign. For example, −2x < 8 becomes x > −4 after dividing both sides by −2. Check with a number: x = 0 gives −2(0) = 0, which is less than 8, and 0 is greater than −4.
The questions start easy and get harder. Try each one before you look at the explanation.
Practice questions Each question and explanation was reviewed by a person. Pick an answer to see if you're right.
Question 1 The formula
T = n + 9 T = n + 9 T = n + 9 gives a total
T T T from a starting number
n n n . Solve the equation for
n n n .
A n = T − 9 n = T - 9 n = T − 9 B n = T + 9 n = T + 9 n = T + 9 C n = 9 − T n = 9 - T n = 9 − T D n = T 9 n = \dfrac{T}{9} n = 9 T E n = 9 T n = 9T n = 9 T
The variable
n n n has
9 9 9 added to it. Undo that by subtracting
9 9 9 from both sides:
n = T − 9 n = T - 9 n = T − 9 . Check with numbers: if
n = 5 n = 5 n = 5 , then
T = 14 T = 14 T = 14 , and
T − 9 = 14 − 9 = 5 T - 9 = 14 - 9 = 5 T − 9 = 14 − 9 = 5 . It matches.
Question 2 Which of the following describes all solutions of the inequality
n + 9 ≤ 4 n + 9 \le 4 n + 9 ≤ 4 ?
A n ≤ 13 n \le 13 n ≤ 13 B n ≤ 5 n \le 5 n ≤ 5 C n ≥ − 5 n \ge -5 n ≥ − 5 D n ≤ − 5 n \le -5 n ≤ − 5 E n ≤ − 13 n \le -13 n ≤ − 13
Subtract 9 from both sides:
n ≤ 4 − 9 n \le 4 - 9 n ≤ 4 − 9 , so
n ≤ − 5 n \le -5 n ≤ − 5 . Subtracting a number from both sides never flips the inequality symbol. Sanity check: try
n = − 6 n = -6 n = − 6 , which satisfies
n ≤ − 5 n \le -5 n ≤ − 5 :
− 6 + 9 = 3 ≤ 4 -6 + 9 = 3 \le 4 − 6 + 9 = 3 ≤ 4 works; try
n = 0 n = 0 n = 0 , which does not:
0 + 9 = 9 0 + 9 = 9 0 + 9 = 9 is not
≤ 4 \le 4 ≤ 4 . So the solution is
n ≤ − 5 n \le -5 n ≤ − 5 .
Question 3 Solve for
x x x :
6 ( x − 2 ) = 2 x + 3 ( x + 1 ) 6(x - 2) = 2x + 3(x + 1) 6 ( x − 2 ) = 2 x + 3 ( x + 1 ) .
A 5 B 9 C 13 D 15 E 17
Distribute both sides:
6 x − 12 = 2 x + 3 x + 3 = 5 x + 3 6x - 12 = 2x + 3x + 3 = 5x + 3 6 x − 12 = 2 x + 3 x + 3 = 5 x + 3 . Subtract
5 x 5x 5 x :
x − 12 = 3 x - 12 = 3 x − 12 = 3 , so
x = 15 x = 15 x = 15 . Check:
6 ( 13 ) = 78 6(13) = 78 6 ( 13 ) = 78 and
2 ( 15 ) + 3 ( 16 ) = 30 + 48 = 78 2(15) + 3(16) = 30 + 48 = 78 2 ( 15 ) + 3 ( 16 ) = 30 + 48 = 78 .
Question 4 Solve the inequality
− 2 ( x − 6 ) ≥ 4 x − 6 -2(x - 6) \ge 4x - 6 − 2 ( x − 6 ) ≥ 4 x − 6 . Which of the following describes all of its solutions?
A x ≥ 3 x \ge 3 x ≥ 3 B x ≤ 9 x \le 9 x ≤ 9 C x ≤ 3 x \le 3 x ≤ 3 D x ≤ − 1 x \le -1 x ≤ − 1 E x ≤ 0 x \le 0 x ≤ 0
Distribute the left side:
− 2 ( x − 6 ) = − 2 x + 12 -2(x-6) = -2x + 12 − 2 ( x − 6 ) = − 2 x + 12 , so
− 2 x + 12 ≥ 4 x − 6 -2x + 12 \ge 4x - 6 − 2 x + 12 ≥ 4 x − 6 . Add
2 x 2x 2 x to both sides and add
6 6 6 :
18 ≥ 6 x 18 \ge 6x 18 ≥ 6 x . Divide by
6 6 6 :
3 ≥ x 3 \ge x 3 ≥ x , i.e.
x ≤ 3 x \le 3 x ≤ 3 . (Check x=3:
− 2 ( − 3 ) = 6 -2(-3)=6 − 2 ( − 3 ) = 6 and
4 ( 3 ) − 6 = 6 4(3)-6=6 4 ( 3 ) − 6 = 6 , and
6 ≥ 6 6 \ge 6 6 ≥ 6 is true; x=4 gives
4 ≥ 10 4 \ge 10 4 ≥ 10 , false.)
Question 5 Which statement best describes the solution set of
3 ( 2 x − 4 ) + 5 = 2 ( 3 x − 1 ) − 5 3(2x - 4) + 5 = 2(3x - 1) - 5 3 ( 2 x − 4 ) + 5 = 2 ( 3 x − 1 ) − 5 ?
A No solution B Infinitely many solutions C Exactly one solution, x = − 7 x = -7 x = − 7 D Exactly one solution, x = 0 x = 0 x = 0
Simplify each side: left is
6 x − 12 + 5 = 6 x − 7 6x - 12 + 5 = 6x - 7 6 x − 12 + 5 = 6 x − 7 ; right is
6 x − 2 − 5 = 6 x − 7 6x - 2 - 5 = 6x - 7 6 x − 2 − 5 = 6 x − 7 . The equation becomes
6 x − 7 = 6 x − 7 6x - 7 = 6x - 7 6 x − 7 = 6 x − 7 , which is true for every value of
x x x , so there are infinitely many solutions.
Question 6 Priya has
$ 30 \$30 $30 saved and adds
$ 8 \$8 $8 to her savings each week. Marco has
$ 54 \$54 $54 saved and adds
$ 5 \$5 $5 to his savings each week. When the two savings amounts are equal, how much money, in dollars, will each of them have?
A $ 86 \$86 $86 B $ 64 \$64 $64 C $ 94 \$94 $94 D $ 118 \$118 $118 E $ 8 \$8 $8
Model each account as a linear function of the number of weeks
w w w : Priya has
30 + 8 w 30 + 8w 30 + 8 w and Marco has
54 + 5 w 54 + 5w 54 + 5 w . Set them equal to find when the amounts match:
30 + 8 w = 54 + 5 w 30 + 8w = 54 + 5w 30 + 8 w = 54 + 5 w , so
3 w = 24 3w = 24 3 w = 24 and
w = 8 w = 8 w = 8 weeks. The question asks for the amount, not the week, so substitute back:
30 + 8 ( 8 ) = 30 + 64 = 94 30 + 8(8) = 30 + 64 = 94 30 + 8 ( 8 ) = 30 + 64 = 94 . Second method: Marco starts
$ 24 \$24 $24 ahead, and Priya closes the gap by
$ 3 \$3 $3 each week, so she catches up in
24 ÷ 3 = 8 24 \div 3 = 8 24 ÷ 3 = 8 weeks; at that time Marco has
54 + 5 ( 8 ) = 54 + 40 = 94 54 + 5(8) = 54 + 40 = 94 54 + 5 ( 8 ) = 54 + 40 = 94 . Both accounts give
$ 94 \$94 $94 , as they must at the equal-value point. Sanity check: the two expressions agree,
30 + 64 = 94 = 54 + 40 30 + 64 = 94 = 54 + 40 30 + 64 = 94 = 54 + 40 . Confirmed.
For parents
These practice questions come from our SHSAT, TACHS and SSAT practice apps. Each app has more than 1,200 practice questions, every one with a worked explanation.
Checked against official sources. How we make this site Wednesday, September 30, 2026