GED Math Linear Inequalities Summary

1. Divide both sides by −3.
2. Because you divide by a negative number, flip the sign: ≤ becomes ≥.
3. 12 ÷ (−3) = −4, so x ≥ −4.

Check: Try x = 0: −3(0) = 0, and 0 ≤ 12 is true. 0 is greater than −4, so it fits x ≥ −4.

Wrong choices

• x ≤ −4: x ≤ −4 forgets to flip the sign; x = −5 gives 15, which is not ≤ 12.
• x ≥ 4: x ≥ 4 drops the negative sign from the answer; x = 0 works but is not ≥ 4.
• x ≤ 4: x ≤ 4 has the wrong number and the wrong direction.

1. Subtract 5 from both sides: 2x > 12.
2. Divide both sides by 2: x > 6.

Check: Try x = 7: 2(7) + 5 = 19, which is greater than 17. Try x = 6: 17 is not greater than 17, so 6 itself is not included.

Wrong choices

• x < 6: x < 6 flips the sign, but you only flip when multiplying or dividing by a negative number.
• x > 11: x > 11 adds 5 instead of subtracting it: (17 + 5) ÷ 2.
• x > 12: x > 12 forgets to divide by 2.

1. Total weight = worker + boxes = 180 + 40b.
2. "At most 2,000" means the total can be 2,000 or less: ≤ 2,000.
3. Inequality: 180 + 40b ≤ 2,000.

Check: With 45 boxes the total is exactly 180 + 1,800 = 1,980 pounds, which is allowed.

Wrong choices

• 180 + 40b ≥ 2,000: ≥ means "at least," which would require the load to be 2,000 pounds or more.
• 180 + 40b < 2,000: < leaves out exactly 2,000 pounds, but "at most" allows 2,000.
• 40 + 180b ≤ 2,000: 40 + 180b switches the numbers, as if each box weighed 180 pounds.

1. Inequality: 45 + 7g ≤ 150.
2. Subtract 45: 7g ≤ 105.
3. Divide by 7: g ≤ 15.
4. The greatest number of guests is 15.

Check: 45 + 7 × 15 = 45 + 105 = $150, exactly the budget.

Wrong choices

• 21: 21 divides $150 by $7 and forgets the $45 room fee.
• 27: 27 adds the $45 fee to the budget instead of subtracting it.
• 16: 16 guests cost 45 + 112 = $157, which is over the $150 budget.

1. A closed circle means 3 is included, so the sign has an "or equal to" part.
2. Shading to the left means numbers less than 3.
3. Together: x ≤ 3.

Check: 3 and every number below 3 (such as 0 or −2) are shaded, and x ≤ 3 includes all of them.

Wrong choices

• x < 3: x < 3 would use an open circle, because 3 is not included.
• x ≥ 3: x ≥ 3 would shade to the right.
• x > 3: x > 3 would use an open circle and shade to the right.

1. Subtract 3x from both sides: 2x − 7 ≥ 9.
2. Add 7 to both sides: 2x ≥ 16.
3. Divide by 2 (a positive number, so no flip): x ≥ 8.

Check: Try x = 8: 5(8) − 7 = 33 and 3(8) + 9 = 33, and 33 ≥ 33 is true.

Wrong choices

• x ≥ 1: x ≥ 1 subtracts 7 from 9 instead of adding 7 to 9.
• x ≤ 8: x ≤ 8 flips the sign, but 2 is positive, so no flip is needed.
• x ≥ 16: x ≥ 16 forgets to divide by 2.

1. Subtract 4: −2x < 6.
2. Divide by −2 and flip the sign: x > −3.
3. Of the values given, only −2 is greater than −3.

Check: 4 − 2(−2) = 4 + 4 = 8, and 8 < 10 is true.

Wrong choices

• x = −3: x = −3 gives 4 + 6 = 10, and 10 < 10 is false.
• x = −4: x = −4 gives 4 + 8 = 12, which is not less than 10.
• x = −5: x = −5 gives 4 + 10 = 14, which is not less than 10.

1. Inequality: 500 + 0.08s ≥ 1,300.
2. Subtract 500: 0.08s ≥ 800.
3. Divide by 0.08: s ≥ 10,000.
4. She needs at least $10,000 in sales.

Check: 8% of $10,000 is $800, and $500 + $800 = $1,300.

Wrong choices

• $16,250: $16,250 divides the full $1,300 by 0.08 and forgets the $500 base pay.
• $6,250: $6,250 divides only the $500 base pay by 0.08.
• $1,000: $1,000 divides 800 by 0.8, which is 80%, not 8%.

1. Total needed: 85 × 4 = 340.
2. Points so far: 78 + 85 + 90 = 253.
3. Fourth test: 340 − 253 = 87, so she needs at least 87.

Check: (78 + 85 + 90 + 87) ÷ 4 = 340 ÷ 4 = 85.

Wrong choices

• 85: 85 gives a total of 338 and an average of 84.5, which is below 85.
• 90: 90 is enough, but it is not the lowest score that works.
• 82: 82 gives a total of 335 and an average of 83.75.

1. Subtract 4: −x/3 > 2.
2. Multiply both sides by −3 and flip the sign: x < −6.

Check: Try x = −9: −(−9)/3 + 4 = 3 + 4 = 7, and 7 > 6 is true. −9 is less than −6, so it fits.

Wrong choices

• x > −6: x > −6 forgets to flip the sign; x = 0 fits it but gives 4, which is not > 6.
• x < 6: x < 6 loses the negative sign; x = 0 fits it but does not work.
• x > 6: x > 6 has both the wrong sign on 6 and the wrong direction; x = 9 gives 1, not more than 6.