Expressions in Action
Evaluate 4x + 3 when x = 5.
23. Substitute: 4(5) + 3 = 20 + 3 = 23.
Simplify x⁴ × x³.
x⁷. Add the exponents because the bases match: 4 + 3 = 7.
Solve 5x - 7 = 18.
x = 5. Add 7 to get 5x = 25, then divide by 5. Check: 5(5) - 7 = 18.
Solve 3x + 2 ≤ 14. Write the answer in interval notation.
x ≤ 4, or (-∞, 4]. Subtract 2, then divide by 3.
For y = -3x + 5, state the slope and the y-intercept as an ordered pair.
Slope = -3. The y-intercept is (0, 5).
Simplify 3(2x - 5) + 4x + 2.
10x - 13. Distribute to get 6x - 15 + 4x + 2, then combine like terms.
Simplify (18a⁶b⁴)/(6a²b). Assume a and b are nonzero.
3a⁴b³. Divide coefficients, then subtract exponents: 18/6 = 3, 6 - 2 = 4, and 4 - 1 = 3.
Solve 4(x - 3) + 5 = 2x + 9.
x = 8. Expand: 4x - 7 = 2x + 9. Then 2x = 16. Check: both original sides equal 25.
Solve 7 - 4x > 19. Write the answer in interval notation.
x < -3, or (-∞, -3). Subtract 7 to get -4x > 12. Divide by -4 and reverse the sign.
Find the slope through (-2, 7) and (4, -5). State whether the line increases or decreases.
m = (-5 - 7)/(4 - (-2)) = -12/6 = -2. The line decreases from left to right.
Evaluate 2a² - 3(a - b) + b² when a = -3 and b = 2. Show the substitution step.
37. Substitute: 2(-3)² - 3(-3 - 2) + 2² = 18 + 15 + 4 = 37.
Simplify [(-2x³y)² × 3xy²]/(6x²y). Use positive exponents. Assume x and y are nonzero.
2x⁵y³. First square: 4x⁶y². Multiply: 12x⁷y⁴. Divide by 6x²y to get 2x⁵y³.
Solve 3(2x - 5) - 2(x + 4) = 5x - 29. Check your answer.
x = 6. Expand: 6x - 15 - 2x - 8 = 5x - 29. Then 4x - 23 = 5x - 29, so x = 6. Check: both original sides equal 1.
Solve 5 - 2(3x - 4) ≥ 4x + 7. Give interval notation and describe the graph.
x ≤ 3/5, or (-∞, 3/5]. Expand: 13 - 6x ≥ 4x + 7. Then 6 ≥ 10x. Use a closed circle at 3/5 and shade left.
Find the equation of the line with slope -3/2 through (-4, 7). Then find y when x = 6.
y = -(3/2)x + 1, and y = -8 when x = 6. Substitute the point: 7 = (-3/2)(-4) + b = 6 + b, so b = 1.
Simplify -3[2(x - 4) - 5] + 2(3x - 1). Explain why the result does not depend on x.
37. Inside the brackets: 2x - 13. Then -6x + 39 + 6x - 2 = 37. The x-terms cancel, leaving a constant.
Simplify (12a⁻²b³)/(3ab⁻¹). Use positive exponents. Then evaluate when a = 2 and b = -1.
4b⁴/a³, with value 1/2. Divide coefficients and subtract exponents: 4a⁻³b⁴ = 4b⁴/a³. Substitute: 4(-1)⁴/2³ = 4/8 = 1/2.
Solve (3x - 2)/4 - (x + 1)/3 = 2. Show how you clear the fractions.
x = 34/5. Multiply both sides by 12: 3(3x - 2) - 4(x + 1) = 24. Then 9x - 6 - 4x - 4 = 24, so 5x = 34. Check: 23/5 - 13/5 = 2.
Solve -5 < 7 - 3x ≤ 16. Give interval notation and explain which endpoints are included.
-3 ≤ x < 4, or [-3, 4). Subtract 7: -12 < -3x ≤ 9. Divide all three parts by -3 and reverse both signs: 4 > x ≥ -3. Include -3 and exclude 4.
A line passes through (-3, 8) and (5, -4). Find its equation and its x-intercept as an ordered pair.
Slope = (-4 - 8)/(5 - (-3)) = -12/8 = -3/2. Using (-3, 8), b = 7/2. Equation: y = -(3/2)x + 7/2. Set y = 0 to get x = 7/3. The x-intercept is (7/3, 0).
A club sells 8 boxes containing n notebooks each for $3 per notebook. It pays $42 in costs and splits the remaining money equally among 6 teams. Write and simplify the amount per team, then evaluate it for n = 7.
Amount per team = (24n - 42)/6 = 4n - 7 dollars. For n = 7, each team receives 4(7) - 7 = $21. Check: total revenue $168, remaining $126, and $126/6 = $21.
Evaluate 27^(2/3) + 16^(3/4) - 2⁻². Give an exact answer and show how each exponent works.
67/4, or 16 3/4. 27^(2/3) = (cube root of 27)² = 9. 16^(3/4) = (fourth root of 16)³ = 8. 2⁻² = 1/4. Therefore 9 + 8 - 1/4 = 67/4.
Plan A charges $18 plus $4 per visit. Plan B charges $42 plus $2.50 per visit. Write and solve an equation for the number of visits at which the plans cost the same. Find that cost and state which plan is cheaper for 20 visits.
18 + 4v = 42 + 2.5v. Thus 1.5v = 24 and v = 16 visits. Both cost $82. At 20 visits, A costs $98 and B costs $92, so Plan B is cheaper by $6.
Solve |3x - 2| ≥ 10. Write interval notation, then list every integer solution from -5 through 5 inclusive.
3x - 2 ≤ -10 OR 3x - 2 ≥ 10. Thus x ≤ -8/3 OR x ≥ 4. Interval: (-∞, -8/3] ∪ [4, ∞). The integers from -5 through 5 are -5, -4, -3, 4, and 5.
A tank drains at a constant rate. It contains 146 liters after 3 minutes and 98 liters after 7 minutes. Write V = mt + b, find when the tank is empty, and state the physically meaningful time interval.
m = (98 - 146)/(7 - 3) = -12 liters/minute. From 146 = -12(3) + b, b = 182 liters. Thus V = -12t + 182. Empty at t = 182/12 = 91/6 minutes, or 15 minutes 10 seconds. Physical domain: 0 ≤ t ≤ 91/6.