π How to Add Integers in algebraic expressions (21 MCQs)
π From Digital SAT Algebra β’ 1. Basics of Algebra β’ 21 questions available
What is How to Add Integers in algebraic expressions?
Definition:
Adding integers in algebraic expressions means performing addition on signed numbers (positive and negative integers) that appear as constants or coefficients within expressions, using the rules of integer addition to combine like terms or evaluate the expression for given variable values.
Working:
When adding integers, if they have the same sign, add their absolute values and keep the sign; if they have different signs, subtract the smaller absolute value from the larger and use the sign of the larger; in expressions, combine integer constants separately from variable terms.
Example:
Simplify .
Solution: Add variable terms: ; add constants: ; result .
Reason:
Mastering integer addition in expressions is necessary for simplifying polynomials, solving linear equations, and working with functions, as it ensures accuracy in operations that involve both positive and negative quantities.
π All How to Add Integers in algebraic expressions MCQs
Q1. A student evaluates by first combining the negative integers and then adding the positive integer. Which result should the student obtain?
π Explanation: The negative integers are and , whose sum is . Adding gives . This checks the meaning of integer addition rather than relying on a memorized sign rule.
Q2. Which expression has the same value as and best demonstrates why the answer is positive?
π Explanation: The two negative integers combine to , while the positive integer is . Therefore, , and also equals . The other choices change the signs or grouping.
Q3. A submarine is meters below sea level. It rises meters, then descends meters, and finally rises meters. What is its final position relative to sea level?
π Explanation: Representing the movements with integers gives . Combining step by step produces , so the submarine finishes meters below sea level. This models each movement using its direction.
Q4. A student claims that because both numbers should be added and the negative sign makes the result negative. What is the best analysis of the error?
π Explanation: The numbers have opposite signs, so their absolute values are compared and the smaller magnitude is subtracted from the larger: . The sign belongs to the number with greater magnitude, so the result is , not .
Q5. On a number line, a point starts at . It moves units to the right and then units to the left. At which integer does it finish?
π Explanation: Moving right represents adding a positive integer, while moving left represents adding a negative integer. Thus the final position is . The direction of each movement is essential for interpreting the graph correctly.
Q6. Two methods are used to evaluate . Method A combines the negative numbers first, while Method B combines with first. Which conclusion is correct?
π Explanation: Method A gives . Method B gives . Both methods preserve the same total, showing that integers can be regrouped without changing their sum.
Q7. Find the integer if . A student argues that because and . Which value of is actually correct?
π Explanation: The equation is , so . Adding to both sides gives . The student's error comes from subtracting the target value instead of undoing both negative additions systematically.
Q8. A student has positive counters and negative counters. After pairing opposite counters, what integer is represented by the remaining counters?
π Explanation: Each positive counter cancels one negative counter. Six positive and six negative counters form six zero pairs, leaving three negative counters. Therefore, the remaining value is . This demonstrates how counters represent both magnitude and sign.
Q9. A model contains positive counters and negative counters. Another student adds negative counters to the model. Which expression and result correctly describe the new model?
π Explanation: The five negative counters represent , and adding three more negative counters represents . Thus the expression is . All eight positive counters can be paired with eight negative counters, leaving no counters.
Q10. A learner represents with seven negative counters and four positive counters. After removing zero pairs, the learner says the answer is . What mistake was made?
π Explanation: A positive counter and a negative counter form a zero pair, so four such pairs can be removed. The remaining three counters are negative, giving . The error was counting all counters instead of recognizing cancellation.
Q11. A game uses positive counters for points earned and negative counters for points lost. Maya has positive counters and negative counters. She then loses more points. How should the counters be changed, and what is the final value?
π Explanation: The initial model represents . Losing more points means adding eight negative counters, giving . This scenario requires interpreting the meaning of a negative quantity before performing the addition.
Q12. A counter diagram is shown conceptually from left to right as positive counters, negative counters, and positive counters. If all possible zero pairs are removed, which value remains?
π Explanation: The five positive and five negative counters cancel completely because each opposite pair has total value zero. The two additional positive counters cannot be canceled, so the model represents . The important step is identifying zero pairs before counting.
Q13. Two students model . Student A creates nine negative counters and six positive counters, then removes six zero pairs. Student B creates nine negative counters and adds six positive counters without removing pairs. Which statement is most accurate?
π Explanation: Both students begin with the same nine negative and six positive counters, so both models represent . Student A simplifies the physical model by removing six zero pairs, while Student B leaves equivalent zero pairs visible. Removing zero pairs changes appearance, not value.
Q14. A student claims that any collection containing equal numbers of positive and negative counters must have a value of , because one positive and one negative counter are always left together as a pair. Which example most strongly disproves the claim?
π Explanation: Five positive counters and five negative counters can be arranged into five zero pairs, so their total value is , not . The example directly disproves the claim and reinforces that opposite counters cancel completely rather than leaving a leftover pair.
Q15. Which expression has a positive result even though it contains both positive and negative integers?
π Explanation: When integers have different signs, compare their absolute values and subtract the smaller from the larger. Since is greater than , , which is positive. The other choices produce negative results.
Q16. A student evaluates as because the student subtracts the smaller absolute value from the larger and chooses the sign of the larger number. What is the correct result?
π Explanation: Both integers are negative, so their absolute values must be added rather than subtracted. Thus , and the result keeps the negative sign. Therefore, . The student's method incorrectly treats same-sign integers as different-sign integers.
Q17. A bank account changes by -\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '' in math mode at position 3: 35\Μ²)Μ², then +\" style="color:#cc0000">35, then , and then . What is the overall change in the account balance?
π Explanation: The changes can be represented as . Combining the negative changes gives , and adding gives . Therefore, the account experiences an overall decrease of .
Q18. A student says because the negative sign should remain and the absolute values should always be added. Which reasoning correctly identifies the error?
π Explanation: The integers have opposite signs, so their absolute values are compared: . Because has the greater absolute value and is negative, the result is . Adding absolute values would incorrectly give .
Q19. A point starts at on a number line, moves units right, and then moves units left. Which statement correctly describes its final position?
π Explanation: Moving right represents , while moving left represents . The expression is . The two movements have a net effect of units right, so the final position is .
Q20. Three students simplify . Student A gets , Student B gets , and Student C gets . Which student is correct, and why?
π Explanation: The expression can be evaluated by combining the positive and negative amounts: . Student A ignores the negative sign of , while Student C adds all absolute values. Only Student B correctly accounts for the signs.
Q21. A contest problem requires three integers to be added: , , and is negative. The final sum must be . Which value of satisfies the condition?
π Explanation: First combine the known integers: . To make the total , the remaining integer must contribute . Thus , giving . This requires reasoning backward from the required sum.