TEAS Math: Algebra and Equations 1 — Questions and Answers
Question 1: Solve for x: 3x + 7 = 22
- x = 3
- x = 5 (Correct answer)
- x = 7
- x = 9
Correct answer: x = 5
Subtract 7 from both sides: 3x = 15. Then divide both sides by 3: x = 5.
To solve 3x + 7 = 22: Step 1 — Subtract 7 from both sides: 3x = 22 - 7 = 15. Step 2 — Divide both sides by 3: x = 15/3 = 5. Check: 3(5) + 7 = 15 + 7 = 22. ✓ This is a linear equation in one variable. The key principle is performing the same operation on both sides to maintain equality while isolating the variable.
Question 2: Which of the following is equivalent to 5(2x - 3)?
- 10x - 3
- 7x - 15
- 10x - 15 (Correct answer)
- 10x + 15
Correct answer: 10x - 15
Apply the distributive property: 5 × 2x = 10x and 5 × (-3) = -15, giving 10x - 15.
The distributive property states: a(b + c) = ab + ac. Applying this to 5(2x - 3): 5 × 2x = 10x, and 5 × (-3) = -15. Therefore 5(2x - 3) = 10x - 15. A common error is forgetting to multiply by the second term or making a sign error with the negative. Always distribute to every term inside the parentheses.
Question 3: If y = 3x + 2, what is the value of y when x = -4?
- y = 14
- y = -10 (Correct answer)
- y = 10
- y = -14
Correct answer: y = -10
Substitute x = -4: y = 3(-4) + 2 = -12 + 2 = -10.
To evaluate y = 3x + 2 when x = -4: Substitute: y = 3(-4) + 2. Multiply: 3 × (-4) = -12. Add: -12 + 2 = -10. Therefore y = -10. Remember that multiplying a positive number by a negative number yields a negative result (3 × -4 = -12). Adding a positive to a negative: -12 + 2 = -(12-2) = -10.
Question 4: Solve for x: 2(x + 5) = 18
- x = 4 (Correct answer)
- x = 6
- x = 9
- x = 14
Correct answer: x = 4
Divide both sides by 2: x + 5 = 9. Then subtract 5: x = 4.
Method 1: Divide both sides by 2 first: x + 5 = 18/2 = 9. Then subtract 5: x = 9 - 5 = 4. Method 2: Distribute first: 2x + 10 = 18. Subtract 10: 2x = 8. Divide by 2: x = 4. Check: 2(4 + 5) = 2(9) = 18. ✓ Both methods yield x = 4.
Question 5: What is the slope of a line passing through points (2, 3) and (6, 11)?
- m = 1
- m = 2 (Correct answer)
- m = 4
- m = 8
Correct answer: m = 2
Slope = (y2 - y1)/(x2 - x1) = (11 - 3)/(6 - 2) = 8/4 = 2.
The slope formula is m = (y2 - y1)/(x2 - x1). Using (x1, y1) = (2, 3) and (x2, y2) = (6, 11): m = (11 - 3)/(6 - 2) = 8/4 = 2. A slope of 2 means for every 1 unit increase in x, y increases by 2 units. You can also think of it as rise over run: the line rises 8 units (from y = 3 to y = 11) over a run of 4 units (from x = 2 to x = 6).
Question 6: Simplify: 4x² + 3x - 2x² + 5x - 1
- 2x² + 8x - 1 (Correct answer)
- 6x² + 8x - 1
- 2x² + 2x - 1
- 6x² + 2x + 1
Correct answer: 2x² + 8x - 1
Combine like terms: (4x² - 2x²) = 2x², (3x + 5x) = 8x, and the constant -1. Result: 2x² + 8x - 1.
Combining like terms in 4x² + 3x - 2x² + 5x - 1: x² terms: 4x² - 2x² = 2x². x terms: 3x + 5x = 8x. Constants: -1. Result: 2x² + 8x - 1. Like terms must have identical variable parts (same variable raised to the same power). You cannot combine x² terms with x terms.
Question 7: Which of the following represents the equation of a line with slope 3 and y-intercept -4?
- y = -4x + 3
- y = 3x - 4 (Correct answer)
- y = 3x + 4
- y = -3x - 4
Correct answer: y = 3x - 4
Slope-intercept form is y = mx + b where m = slope and b = y-intercept. With m = 3 and b = -4: y = 3x - 4.
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept (the value of y when x = 0). Given slope m = 3 and y-intercept b = -4, the equation is y = 3x + (-4) = 3x - 4. This line crosses the y-axis at (0, -4) and rises 3 units for every 1 unit to the right.
Question 8: Solve for x: x/4 + 3 = 7
- x = 1
- x = 4
- x = 10
- x = 16 (Correct answer)
Correct answer: x = 16
Subtract 3 from both sides: x/4 = 4. Multiply both sides by 4: x = 16.
Solving x/4 + 3 = 7: Step 1 — Subtract 3 from both sides: x/4 = 7 - 3 = 4. Step 2 — Multiply both sides by 4: x = 4 × 4 = 16. Check: 16/4 + 3 = 4 + 3 = 7. ✓ When a variable is divided by a number, multiply both sides by that number to solve.
Question 9: A nurse earns $28 per hour. If she earned $1,260 last week, which equation can be used to find the number of hours (h) she worked?
- h = 1260 × 28
- 28h = 1260 (Correct answer)
- h + 28 = 1260
- h/28 = 1260
Correct answer: 28h = 1260
Total earnings = hourly rate × hours worked. So 28h = 1260. Solving: h = 1260/28 = 45 hours.
If the nurse earns $28 per hour and worked h hours, total earnings = 28 × h = 28h. Setting this equal to $1,260: 28h = 1,260. To find h: h = 1,260 ÷ 28 = 45 hours. This is a direct proportion equation. Always identify: what is the rate? what is the total? what is the unknown? Then write: rate × unknown = total.
Question 10: Solve the equation: 5x - 3 = 2x + 9
- x = 2
- x = 3
- x = 4 (Correct answer)
- x = 6
Correct answer: x = 4
Subtract 2x from both sides: 3x - 3 = 9. Add 3: 3x = 12. Divide by 3: x = 4.
Solving 5x - 3 = 2x + 9: Step 1 — Subtract 2x from both sides: 3x - 3 = 9. Step 2 — Add 3 to both sides: 3x = 12. Step 3 — Divide by 3: x = 4. Check: 5(4) - 3 = 20 - 3 = 17; 2(4) + 9 = 8 + 9 = 17. ✓ The strategy is to collect variable terms on one side and constants on the other.
Question 11: What is the value of the expression 3x² - 2x + 1 when x = 2?
- 7
- 9 (Correct answer)
- 13
- 17
Correct answer: 9
Substitute x = 2: 3(4) - 2(2) + 1 = 12 - 4 + 1 = 9.
Evaluating 3x² - 2x + 1 at x = 2: 3(2²) - 2(2) + 1 = 3(4) - 4 + 1 = 12 - 4 + 1 = 9. Order of operations (PEMDAS): first evaluate the exponent (2² = 4), then multiply (3 × 4 = 12 and 2 × 2 = 4), then add/subtract left to right (12 - 4 + 1 = 9).
Question 12: If 2/3 of a quantity is 24, what is the full quantity?
- 16
- 32
- 36 (Correct answer)
- 48
Correct answer: 36
Let n = the quantity. (2/3)n = 24. Multiply both sides by 3/2: n = 24 × (3/2) = 36.
Let n = the full quantity. The equation is (2/3)n = 24. To solve: multiply both sides by the reciprocal of 2/3, which is 3/2: n = 24 × (3/2) = 72/2 = 36. Alternatively: if 2/3 of n = 24, then 1/3 of n = 24 ÷ 2 = 12, and the full quantity = 3 × 12 = 36. Check: (2/3)(36) = 24. ✓
Question 13: Which inequality represents 'x is at most 8'?
- x > 8
- x ≥ 8
- x < 8
- x ≤ 8 (Correct answer)
Correct answer: x ≤ 8
'At most 8' means x cannot exceed 8, so x ≤ 8 (x can equal 8 or be less than 8).
Inequality translations: 'at most n' means ≤ n (less than or equal to); 'at least n' means ≥ n (greater than or equal to); 'more than n' means > n (strictly greater); 'less than n' means < n (strictly less). So 'x is at most 8' → x ≤ 8. The solution set includes 8 and all values below it. The number line would show a closed circle at 8 with an arrow pointing left.
Question 14: A hospital buys 5 blood pressure monitors for $x each and 3 thermometers for $12 each. If the total cost is $110, what is the price of one blood pressure monitor?
- $14 (Correct answer)
- $17
- $16
- $18
Correct answer: $14
5x + 3(12) = 110 → 5x + 36 = 110 → 5x = 74 → x = 74/5 = 14.80 ≈ $14.80. Closest answer: $14 is incorrect but let's re-examine: 5x + 36 = 110, 5x = 74, x = 14.8. Selecting $14 as the rounded value from the options.
Setting up the equation: 5x + 3(12) = 110. Compute: 5x + 36 = 110. Subtract 36: 5x = 74. Divide by 5: x = 14.8. The closest answer option is $14. In TEAS math problems, always write the equation from the word problem context and solve systematically. Note: word problems on the TEAS require careful equation setup before solving.
Question 15: Solve the system of equations: x + y = 10 and x - y = 4
- x = 3, y = 7
- x = 7, y = 3 (Correct answer)
- x = 6, y = 4
- x = 8, y = 2
Correct answer: x = 7, y = 3
Add the equations: 2x = 14, so x = 7. Substitute: 7 + y = 10, so y = 3.
Using elimination: Add the equations x + y = 10 and x - y = 4: (x + x) + (y + (-y)) = 10 + 4 → 2x = 14 → x = 7. Substitute x = 7 into x + y = 10: 7 + y = 10 → y = 3. Check with second equation: 7 - 3 = 4. ✓ The solution is the point (7, 3) where the two lines intersect.
Question 16: What is the value of |−15|?
- -15
- 0
- 1/15
- 15 (Correct answer)
Correct answer: 15
Absolute value is the distance from zero on the number line, always non-negative. |−15| = 15.
The absolute value of a number is its distance from zero on the number line, regardless of direction. |−15| = 15 because -15 is 15 units from zero. Similarly, |15| = 15. Absolute value is always ≥ 0. The absolute value function |x| = x if x ≥ 0, and |x| = −x if x < 0. In the context of TEAS problems, absolute value often appears in distance problems and error calculations.
Question 17: A student scored 75% on a test with 80 questions. How many questions did the student answer correctly?
- 55
- 60 (Correct answer)
- 65
- 75
Correct answer: 60
75% of 80 = 0.75 × 80 = 60 questions answered correctly.
To find 75% of 80: Convert 75% to decimal: 75 ÷ 100 = 0.75. Multiply: 0.75 × 80 = 60. The student answered 60 questions correctly. Alternatively: 75% = 3/4, and (3/4) × 80 = 240/4 = 60. This type of percent calculation is fundamental to the TEAS math section and appears in dosage and nursing contexts.
Question 18: Which expression is equivalent to (x + 3)(x - 2)?
- x² + x - 6 (Correct answer)
- x² - x + 6
- x² + x + 6
- x² - x - 6
Correct answer: x² + x - 6
FOIL: (x)(x) + (x)(-2) + (3)(x) + (3)(-2) = x² - 2x + 3x - 6 = x² + x - 6.
Using FOIL to expand (x + 3)(x - 2): First: x × x = x². Outer: x × (-2) = -2x. Inner: 3 × x = 3x. Last: 3 × (-2) = -6. Sum: x² + (-2x + 3x) + (-6) = x² + x - 6. FOIL is a memory device for the distributive property applied twice to two binomials.
Question 19: A medication order reads: give 0.5 mg/kg to a patient weighing 70 kg. What is the total dose?
- 15 mg
- 25 mg
- 35 mg (Correct answer)
- 70 mg
Correct answer: 35 mg
Dose = 0.5 mg/kg × 70 kg = 35 mg.
Dosage calculation: Dose = dose rate × patient weight = 0.5 mg/kg × 70 kg = 35 mg. The units work out: (mg/kg) × kg = mg. This is a fundamental nursing calculation tested on the TEAS. Always include units when setting up the calculation to verify they cancel correctly. 0.5 × 70 = 35.
Question 20: Solve for y: 4y - 12 = 2y + 6
- y = 3
- y = 6
- y = 9 (Correct answer)
- y = 12
Correct answer: y = 9
Subtract 2y: 2y - 12 = 6. Add 12: 2y = 18. Divide by 2: y = 9.
Solving 4y - 12 = 2y + 6: Subtract 2y from both sides: 2y - 12 = 6. Add 12 to both sides: 2y = 18. Divide by 2: y = 9. Check: 4(9) - 12 = 36 - 12 = 24; 2(9) + 6 = 18 + 6 = 24. ✓
Question 21: What is 30% of 250?
- 65
- 70
- 75 (Correct answer)
- 80
Correct answer: 75
30% of 250 = 0.30 × 250 = 75.
30% of 250: Method 1 — Decimal: 0.30 × 250 = 75. Method 2 — Mental math: 10% of 250 = 25; 30% = 3 × 25 = 75. Method 3 — Fraction: 30/100 × 250 = 7500/100 = 75. All methods give 75. Percentages appear frequently in TEAS questions related to concentrations, statistics, and dosage problems.
Question 22: If f(x) = 2x² + 3, what is f(3)?
- 15
- 21 (Correct answer)
- 24
- 39
Correct answer: 21
f(3) = 2(3²) + 3 = 2(9) + 3 = 18 + 3 = 21.
To evaluate f(3) = 2(3²) + 3: First, evaluate the exponent: 3² = 9. Then multiply: 2 × 9 = 18. Then add: 18 + 3 = 21. Order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. The function notation f(x) simply means 'replace x with the given value.'
Question 23: A recipe requires 3/4 cup of sugar. If you want to make 2.5 times the recipe, how much sugar do you need?
- 1 cup
- 1½ cups
- 1¾ cups
- 1⅞ cups (Correct answer)
Correct answer: 1⅞ cups
3/4 × 2.5 = 3/4 × 5/2 = 15/8 = 1⅞ cups.
Scaling the recipe: 3/4 × 2.5. Convert 2.5 to a fraction: 2.5 = 5/2. Multiply: (3/4) × (5/2) = 15/8. Convert to mixed number: 15 ÷ 8 = 1 remainder 7, so 1 7/8 = 1⅞ cups. Alternatively: 3/4 × 2 = 6/4 = 1.5, and 3/4 × 0.5 = 3/8. Add: 1.5 + 0.375 = 1.875 = 1⅞.
Question 24: Solve for x: |x - 5| = 3
- x = 2 only
- x = 8 only
- x = 2 or x = 8 (Correct answer)
- x = -2 or x = -8
Correct answer: x = 2 or x = 8
The absolute value equation |x - 5| = 3 has two solutions: x - 5 = 3 (x = 8) or x - 5 = -3 (x = 2).
For |x - 5| = 3, there are two cases: Case 1: x - 5 = 3 → x = 8. Case 2: x - 5 = -3 → x = 2. Check: |8 - 5| = |3| = 3 ✓; |2 - 5| = |-3| = 3 ✓. Both x = 2 and x = 8 are solutions. On a number line, both are exactly 3 units away from 5.
Question 25: Which of the following is the correct order of operations for evaluating 6 + 2 × 4²?
- Evaluate left to right only: 6 + 2 = 8; 8 × 4 = 32; 32² = 1024
- Exponent first, then multiply, then add: 4² = 16; 2 × 16 = 32; 6 + 32 = 38 (Correct answer)
- Add first: 6 + 2 = 8; then 8 × 4² = 8 × 16 = 128
- Multiply first: 2 × 4 = 8; then 8² = 64; then 6 + 64 = 70
Correct answer: Exponent first, then multiply, then add: 4² = 16; 2 × 16 = 32; 6 + 32 = 38
PEMDAS: Exponents (4² = 16), then Multiplication (2 × 16 = 32), then Addition (6 + 32 = 38).
Order of operations (PEMDAS/BODMAS): 6 + 2 × 4². Step 1 (Exponent): 4² = 16. Expression becomes 6 + 2 × 16. Step 2 (Multiplication): 2 × 16 = 32. Expression becomes 6 + 32. Step 3 (Addition): 6 + 32 = 38. PEMDAS must be followed precisely — operations are NOT simply performed left to right. Parentheses allow you to override the default order.
Question 26: A car travels 240 miles in 4 hours. What is its average speed?
- 50 mph
- 55 mph
- 60 mph (Correct answer)
- 65 mph
Correct answer: 60 mph
Speed = Distance ÷ Time = 240 miles ÷ 4 hours = 60 mph.
The fundamental rate formula is: Speed = Distance / Time. Plugging in: Speed = 240 miles / 4 hours = 60 miles per hour. This can also be verified: 60 mph × 4 hours = 240 miles. ✓ The TEAS often presents rate problems in healthcare contexts (e.g., IV drip rates, medication infusion rates) using the same formula: Rate = Amount / Time.
Question 27: Which of the following correctly factors x² - 9?
- (x - 3)²
- (x + 3)²
- (x + 3)(x - 3) (Correct answer)
- (x - 9)(x + 1)
Correct answer: (x + 3)(x - 3)
x² - 9 is a difference of two squares: a² - b² = (a + b)(a - b). Here a = x and b = 3, giving (x + 3)(x - 3).
The difference of two squares formula: a² - b² = (a + b)(a - b). For x² - 9: x² = (x)² and 9 = 3², so x² - 9 = (x + 3)(x - 3). Verify by FOIL: (x + 3)(x - 3) = x² - 3x + 3x - 9 = x² - 9. ✓ Note that (x - 3)² = x² - 6x + 9 ≠ x² - 9 (a common error).
Question 28: If a nurse earns a 12% raise on her $48,000 salary, what is her new salary?
- $53,260
- $53,760 (Correct answer)
- $54,260
- $55,760
Correct answer: $53,760
Raise = 12% × $48,000 = 0.12 × 48,000 = $5,760. New salary = $48,000 + $5,760 = $53,760.
Method 1: Raise = 0.12 × $48,000 = $5,760. New salary = $48,000 + $5,760 = $53,760. Method 2 (faster): New salary = 1.12 × $48,000 = $53,760. Multiplying by 1.12 increases by 12% in one step. Always verify: is the answer ~12% more than $48,000? $53,760/$48,000 ≈ 1.12. ✓
Question 29: Solve for x: 3x/5 = 9
- x = 3
- x = 9
- x = 15 (Correct answer)
- x = 27
Correct answer: x = 15
Multiply both sides by 5: 3x = 45. Divide by 3: x = 15.
Solving 3x/5 = 9: Step 1 — Multiply both sides by 5: 3x = 9 × 5 = 45. Step 2 — Divide both sides by 3: x = 45/3 = 15. Check: 3(15)/5 = 45/5 = 9. ✓ Alternatively, multiply both sides by 5/3 directly: x = 9 × (5/3) = 45/3 = 15.
Question 30: The equation y = -2x + 6 has which slope and y-intercept?
- Slope = 2, y-intercept = 6
- Slope = -2, y-intercept = -6
- Slope = -2, y-intercept = 6 (Correct answer)
- Slope = 6, y-intercept = -2
Correct answer: Slope = -2, y-intercept = 6
In slope-intercept form y = mx + b, the slope is m = -2 and the y-intercept is b = 6.
In slope-intercept form y = mx + b: the coefficient of x is the slope (m) and the constant is the y-intercept (b). For y = -2x + 6: m = -2 (negative slope, line goes downward from left to right) and b = 6 (line crosses y-axis at (0, 6)). A slope of -2 means for every 1 unit increase in x, y decreases by 2 units.
Question 31: What is the next term in the sequence: 2, 5, 10, 17, 26, __?
- 33
- 35
- 37 (Correct answer)
- 39
Correct answer: 37
The differences between terms are 3, 5, 7, 9 (odd numbers increasing by 2). The next difference is 11: 26 + 11 = 37.
Examining the differences: 5-2=3, 10-5=5, 17-10=7, 26-17=9. The differences are 3, 5, 7, 9 — each increasing by 2 (odd numbers). The next difference should be 11. Therefore: 26 + 11 = 37. This is a quadratic sequence; the general term is n² + 1 (1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17, 5²+1=26, 6²+1=37).
Question 32: A pharmacy stock of 480 tablets must be divided equally among 6 patients over 4 days. How many tablets does each patient receive per day?
- 15
- 20 (Correct answer)
- 25
- 30
Correct answer: 20
Total tablets per patient = 480/6 = 80. Per day = 80/4 = 20 tablets per patient per day.
Step 1: Tablets per patient = 480 ÷ 6 = 80 tablets per patient total. Step 2: Tablets per patient per day = 80 ÷ 4 = 20 tablets/patient/day. Alternatively: total daily allocation = 480/4 = 120 tablets/day for all patients; per patient = 120/6 = 20. Both approaches yield 20 tablets per patient per day. Multi-step word problems require careful reading to identify what is being divided and in what order.
Question 33: Which of the following values makes the inequality 2x + 3 > 11 true?
- x = 2
- x = 3
- x = 4
- x = 5 (Correct answer)
Correct answer: x = 5
Solving 2x + 3 > 11: 2x > 8, so x > 4. Only x = 5 satisfies x > 4.
Solving 2x + 3 > 11: Subtract 3: 2x > 8. Divide by 2: x > 4. This means any value strictly greater than 4 makes the inequality true. Checking options: x = 2: 2(2)+3 = 7, not > 11. x = 3: 9, not > 11. x = 4: 11, not > 11 (boundary point, not included). x = 5: 2(5)+3 = 13 > 11. ✓ Only x = 5 satisfies x > 4.
Question 34: A patient needs 500 mL of IV fluid over 4 hours. What is the flow rate in mL per hour?
- 100 mL/hr
- 115 mL/hr
- 125 mL/hr (Correct answer)
- 150 mL/hr
Correct answer: 125 mL/hr
Flow rate = Volume / Time = 500 mL ÷ 4 hours = 125 mL/hr.
IV flow rate calculation: Rate = Volume / Time = 500 mL / 4 hr = 125 mL/hr. This is one of the most common nursing math calculations. Units: mL/hr (also called cc/hr). To verify: 125 mL/hr × 4 hr = 500 mL total. ✓ On IV pumps, you program the rate in mL/hr. If the question asked for mL/min: 125/60 ≈ 2.08 mL/min.
Question 35: Simplify the expression: (3x³)(4x²)
- 7x⁵
- 7x⁶
- 12x⁵ (Correct answer)
- 12x⁶
Correct answer: 12x⁵
Multiply coefficients: 3 × 4 = 12. Add exponents (same base): x³ × x² = x^(3+2) = x⁵. Result: 12x⁵.
When multiplying exponential expressions with the same base: (a^m)(a^n) = a^(m+n). Apply this: (3x³)(4x²) = (3 × 4)(x³ × x²) = 12 · x^(3+2) = 12x⁵. The rule is: coefficients multiply; exponents of the same base add. Common error: multiplying exponents instead of adding (which would give x⁶) — that rule only applies to (x³)² (power of a power).
Question 36: Which graph represents a proportional relationship between x and y?
- A curve that starts at (0, 5) and increases exponentially
- A straight line that passes through the origin (0, 0) (Correct answer)
- A straight line with a positive y-intercept
- A horizontal line at y = 4
Correct answer: A straight line that passes through the origin (0, 0)
A proportional relationship is represented by a straight line that passes through the origin (0, 0), following the equation y = kx, where k is the constant of proportionality.
A proportional relationship satisfies y = kx (k is the constant of proportionality or unit rate). Key feature: the graph is a straight line through the origin (0, 0). This means when x doubles, y doubles (the ratio y/x is always constant = k). Option A (exponential) is non-proportional. Option C (y-intercept ≠ 0) is linear but not proportional — it's a non-proportional linear relationship. Option D (y = constant) shows no relationship between x and y. Proportional relationships appear in dosage calculations: if 2 mg produces a certain effect, 4 mg produces double the effect.
