Ans: (a) Explanation: Combine the like terms by adding and subtracting the coefficients: \(3 + 5 - 2 = 6\). Therefore, \(3x + 5x - 2x = 6x\).
Q2: Which of the following represents the equation "Five more than three times a number is 20"? (a) \(5x + 3 = 20\) (b) \(3x + 5 = 20\) (c) \(5 + x + 3 = 20\) (d) \(3 + 5x = 20\)
Solution:
Ans: (b) Explanation: "Three times a number" means \(3x\), and "five more than" means we add 5. So the equation is \(3x + 5 = 20\). Option (a) incorrectly multiplies 5 by \(x\), and options (c) and (d) do not correctly represent the phrase.
Q5: Which inequality represents "A number \(n\) is at least 12"? (a) \(n <> (b) \(n > 12\) (c) \(n \leq 12\) (d) \(n \geq 12\)
Solution:
Ans: (d) Explanation: "At least 12" means the number can be 12 or greater, which is represented by the inequality \(n \geq 12\). Option (c) would mean "at most 12," which is incorrect.
Ans: (a) Explanation: Use the distributive property:
\(2(x + 4) - 3x = 2x + 8 - 3x\)
Combine like terms: \(2x - 3x = -x\)
Final result: \(-x + 8\)
Q8: Which value of \(x\) makes the inequality \(3x + 2 < 11\)=""> (a) \(x = 4\) (b) \(x = 3\) (c) \(x = 2\) (d) \(x = 5\)
Solution:
Ans: (c) Explanation: First, solve the inequality:
\(3x + 2 <>
\(3x <>
\(x <>
Only \(x = 2\) is less than 3. Testing: \(3(2) + 2 = 8 < 11\)="" ✓.="" the="" other="" options="" give="" values="" that="" are="" not="" less="" than="" 3.="">
Section B: Fill in the Blanks
Q9:The result of combining like terms in an expression is called __________ the expression.
Solution:
Ans: simplifying Explanation: When we combine like terms, we are simplifying the expression to make it shorter and easier to work with.
Q10:In the equation \(5x - 8 = 12\), the value of \(x\) is __________.
Solution:
Ans: 4 Explanation: Solve by adding 8 to both sides: \(5x = 20\), then divide by 5: \(x = 4\).
Q11:The property that states \(a(b + c) = ab + ac\) is called the __________ property.
Solution:
Ans: distributive Explanation: The distributive property allows us to multiply a number by a sum by distributing the multiplication to each term inside the parentheses.
Q12:If \(x + 9 = 15\), then \(x = \) __________.
Solution:
Ans: 6 Explanation: Subtract 9 from both sides: \(x = 15 - 9 = 6\).
Q13:The inequality \(x \leq 7\) means that \(x\) is less than or equal to __________.
Solution:
Ans: 7 Explanation: The symbol \(\leq\) represents "less than or equal to," so \(x\) can be any value up to and including 7.
Q14:When solving \(2x = 18\), we divide both sides by __________ to isolate \(x\).
Solution:
Ans: 2 Explanation: To isolate the variable \(x\), we perform the inverse operation of multiplication, which is division. Dividing both sides by 2 gives \(x = 9\).
Section C: Word Problems
Q15:Maria has 23 dollars. She earns \(d\) dollars babysitting. Write an expression for the total amount of money Maria has now. If she earned 15 dollars, how much money does she have in total?
Solution:
Ans:
The expression for total money is \(23 + d\).
Substitute \(d = 15\):
\(23 + 15 = 38\) Final Answer: 38 dollars
Q16:The perimeter of a rectangle is given by the formula \(P = 2l + 2w\), where \(l\) is the length and \(w\) is the width. If the perimeter is 48 cm and the length is 15 cm, find the width.
Solution:
Ans:
Substitute the given values into the formula:
\(48 = 2(15) + 2w\)
\(48 = 30 + 2w\)
Subtract 30 from both sides:
\(18 = 2w\)
Divide by 2:
\(w = 9\) Final Answer: 9 cm
Q17:A cell phone plan costs 25 dollars per month plus 0.10 dollars per text message. Write an equation for the total monthly cost \(C\) if \(t\) text messages are sent. How much will the bill be if 80 text messages are sent?
Solution:
Ans:
The equation is \(C = 25 + 0.10t\).
Substitute \(t = 80\):
\(C = 25 + 0.10(80)\)
\(C = 25 + 8\)
\(C = 33\) Final Answer: 33 dollars
Q18:Jake scored 18 points fewer than twice the number of points Sam scored. If Jake scored 34 points, how many points did Sam score?
Solution:
Ans:
Let \(s\) represent Sam's score.
Jake's score is \(2s - 18\).
Set up the equation: \(2s - 18 = 34\)
Add 18 to both sides:
\(2s = 52\)
Divide by 2:
\(s = 26\) Final Answer: 26 points
Q19:Solve the inequality and describe the solution: A bakery needs to make at least 150 cookies for an order. They have already made 87 cookies. How many more cookies \(c\) do they need to make?
Solution:
Ans:
Set up the inequality: \(87 + c \geq 150\)
Subtract 87 from both sides:
\(c \geq 63\)
The bakery needs to make at least 63 more cookies. Final Answer: At least 63 cookies
Q20:The sum of three consecutive integers is 72. Find the three integers.
Solution:
Ans:
Let the first integer be \(n\).
The three consecutive integers are \(n\), \(n + 1\), and \(n + 2\).
Set up the equation:
\(n + (n + 1) + (n + 2) = 72\)
Combine like terms:
\(3n + 3 = 72\)
Subtract 3 from both sides:
\(3n = 69\)
Divide by 3:
\(n = 23\)
The three integers are 23, 24, and 25. Final Answer: 23, 24, and 25
The document Worksheet (with Solutions): Expressions, Equations, & Inequalities is a part of the Grade 7 Course Math Grade 7.
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