Essential Foundations — Starting check

C; 12. Subtract 2 to get 3x = 18, so x = 6. The target is 2x, not x, and 2(6) = 12. You can also double 3x = 18 to get 6x = 36 and divide by 3. Choice A stops at an intermediate value. Repair: Example 1.01.

3/8 or 0.375. Divide both sides of 8a = 3 by 8. The fraction 3/8 is exact; converting it to a decimal gives 0.375. Both 3/8 and.375 are suitable numeric entries. Reversing the quotient to 8/3 would not satisfy the equation. Check: 8(3/8) = 3. Repair: Example 1.03.

12 new members. Fifteen percent means 15/100 = 0.15, so the number of new members is 0.15(80) = 12. A mental check is 10% of 80 plus 5% of 80: 8 + 4 = 12. The requested answer is a member count, not the percent number 15. Repair: Example 1.09.

A; 1. Multiplication precedes subtraction: 7 − 2(3) = 7 − 6 = 1. Computing (7 − 2)(3) would instead give 15, but there are no parentheses grouping 7 − 2 in the original expression. Read grouping before applying operations. Repair: Example 2.02.

−17. First multiply 4(−2) = −8. Then −9 + (−8) = −17. Adding two negative quantities moves farther below zero; it does not produce a positive sum. A number-line check starts at −9 and moves 8 units left. Repair: Example 2.01.

7/12. Use a common denominator of 12: 5/6 = 10/12 and 1/4 = 3/12. Subtract the numerators to get (10 − 3)/12 = 7/12. Do not subtract denominators. The result is smaller than 5/6 and positive, as subtracting 1/4 should produce. Repair: Example 2.04.

2/3.

Division by a nonzero fraction is multiplication by its reciprocal:

$\frac{\text{3}}{\text{5}}$ ÷ $\frac{\text{9}}{\text{10}}$ = $\frac{\text{3}}{\text{5}}$ · $\frac{\text{10}}{\text{9}}$ = $\frac{\text{30}}{\text{45}}$ = $\frac{\text{2}}{\text{3}}$.

Only the divisor is inverted. Because 9/10 is less than 1, the quotient should be greater than 3/5; 2/3 is. Repair: Example 2.07.

0. The first term is −3² = −(3²) = −9. The second is (−3)² = (−3)(−3) = 9. Their sum is 0. Parentheses determine whether the negative sign belongs to the base that is squared. Repair: Example 2.02.

A; 6.4 × 10⁻³. Moving the decimal in 0.0064 three places to the right produces the coefficient 6.4, so the compensating power is 10⁻³. Multiplying 6.4 by 0.001 restores 0.0064. A positive exponent would make the value large, not small. Repair: Example 2.12.

4.7. Keep the fraction exact until the end: 7(2/3) = 14/3 = 4.666.... The hundredths digit is 6, so rounding to the nearest tenth gives 4.7. Rounding 2/3 first can change the final answer. Repair: Example 2.14.

B; 7x − 12. Distribute 4 to both terms: 4(2x − 3) − x = 8x − 12 − x. Combine the like terms 8x − x = 7x to obtain 7x − 12. The constant stays −12; it is not combined with an x-term. Repair: Example 3.05.

−6. Substitute with parentheses: −2(−3)² + 3(4) = −2(9) + 12 = −18 + 12 = −6. The square belongs to a, not to the entire term −2a. A negative coefficient does not disappear when the variable is squared. Repair: Example 3.02.

−4. The coefficient is the signed number multiplying the variable. In −4x + 9, that number is −4, not 4. The constant term is 9. Reading each term with its preceding sign prevents errors during later combining and factoring. Repair: Example 3.01.

B; 3. Subtract 1 from both sides of 3x + 1 = 10 to get 3x = 9, then divide by 3 to get x = 3. Substitution gives 3(3) + 1 = 10. A solution is a value that makes the original equality true. Repair: Example 3.09.

B; x + 3, with the original restriction x ≠ 3. Factor the numerator: x² − 9 = (x − 3)(x + 3). When x ≠ 3, divide out the nonzero factor x − 3 to get x + 3. The simplified expression alone is defined at 3, but the original quotient is not. The restriction stays. Repair: Example 3.08.

7. The input is −2. Replace every x in f(x) = x² + 3 with (−2): f(−2) = (−2)² + 3 = 4 + 3 = 7. Function notation names an output; it does not mean multiplication by a separate number f. Repair: Example 4.03.

1. The question gives the output 0 and asks for its input. Locate 0 in the g(x) row; the aligned x-entry is 1. This reverses the direction of a question such as “what is g(0)?” Label the requested input before reading across the table. Repair: Example 4.05.

2 units of output per unit of input. The output increases from 5 to 11 while the input increases from 1 to 4. Thus the rate is (11 − 5)/(4 − 1) = 6/3 = 2. An output difference of 6 alone is not a rate because it occurred over 3 input units. Repair: Example 4.07.

B; integers from 0 through 18, inclusive.

The number of occupied passenger seats is a whole-number count. It cannot be negative or exceed 18. Therefore the domain is {0, 1, 2,..., 18}, not every real number in that interval. A count of 2.5 occupied seats is not permitted by the model. Repair: Example 4.10.

C; 3x − 7. Start with “three times x,” which is 3x, then subtract 7 because the result is seven less than that amount. The expression 7 − 3x reverses the comparison. With x = 10, the described amount is seven less than 30, namely 23. Repair: Example 5.01.

24 adults. The ratio has 3 + 2 = 5 equal parts. Forty people correspond to 5 parts, so each part represents 8 people. Adults account for 3 parts: 3(8) = 24. The adult share is 3/5, not 3/2, of the total. Repair: Example 5.04.

9 items. Let n be the nonnegative integer number of items. The budget condition is 6 + 3n ≤ 33, so 3n ≤ 27 and n ≤ 9. Nine items cost exactly $33 and ten cost $36, so 9 is the greatest permitted count. Repair: Example 5.10.

360 meters.

Multiply speed by time: 45$\frac{\text{meters}}{\text{minute}}$ · 8 minutes = 360 meters.

The minute units cancel, leaving the unit asked for. Dividing 45 by 8 would give the wrong quantity and the wrong units. Repair: Example 5.08.