Inequalities Lesson Test

Answer: D

Correct answer: x > 11. Add 7 to both sides:

Adding or subtracting never changes the direction of the sign.

• A: This comes from subtracting 7 instead of adding it to undo the subtraction.
• B: The original sign is strict, so 11 itself is not a solution and the symbol must stay >.
• C: Adding a number never reverses the sign; the direction was flipped for no reason.
• D (correct): Adding 7 to both sides isolates x and keeps the direction, giving x > 11.

Answer: B

Correct answer:

The filled circle shows the boundary −2 is part of the solution, so the sign is ≤ or ≥. Shading to the left means all smaller numbers, so the inequality is

• A: Greater-than values lie to the right of −2, but the shading goes left.
• B (correct): A closed circle means −2 is included, and shading to the left means smaller values, so x ≤ −2.
• C: This would need an open circle and shading to the right, the opposite of the graph.
• D: A strict sign would be drawn with an open circle, not a filled one.

Answer: C

Correct answer:

Divide both sides by −5. Because the divisor is negative, the sign reverses:

Check x = 0: 0 < 30 is true, and

• A: This treats −5 as +5, losing both the negative result and the required flip.
• B: This divides by −5 correctly but forgets to reverse the inequality sign.
• C (correct): Dividing both sides by −5 gives −6 and reverses the sign, so x > −6.
• D: The sign was reversed, but 30 divided by −5 is −6, not 6.

Answer: C

Correct answer: [-1, ∞). The solution starts at −1, which is included because of ≥, so use a bracket. It continues forever to the right, and infinity always gets a parenthesis: $[-1,\infty )$.

• A: A parenthesis at −1 would leave out the boundary, but ≥ includes it.
• B: Infinity is not a number that can be reached, so it never gets a bracket.
• C (correct): A bracket includes −1, and infinity always takes a parenthesis.
• D: This describes values at most −1, the side opposite the solution.

Answer: A

Correct answer: 4. Subtract 3:

Divide by 2:

Only 4 is at most 4; the sign ≤ includes the boundary, so 4 works.

• A (correct): Substituting 4 gives 11 ≤ 11, which is true because the boundary is included.
• B: Substituting 6 gives 15, which is greater than 11, so it fails.
• C: Substituting 4.5 gives 12, which is greater than 11, so it fails.
• D: Substituting 5 gives 13, which is greater than 11, so it fails.

Answer: B

Correct answer:

Distribute:

Subtract 6:

Divide by −2 and reverse the sign:

Check

which is at least 14.

• A: This comes from distributing −2 to −3 as −6 instead of +6, giving −2x ≥ 20.
• B (correct): Distributing gives −2x + 6 ≥ 14, so −2x ≥ 8, and dividing by −2 with a flip gives x ≤ −4.
• C: This multiplies only x by −2 and leaves −3, giving −2x − 3 ≥ 14.
• D: The arithmetic is right, but dividing by −2 requires reversing the sign.

Answer: A

Correct answer: x < 5. Subtract 3x from both sides:

Add 4:

Divide by the positive number 4:

• A (correct): Subtracting 3x and adding 4 gives 4x < 20, so x < 5.
• B: This subtracts 4 from 16 instead of adding 4, giving 4x < 12.
• C: The coefficient 4 is positive, so no flip is needed; reversing the sign was a mistake.
• D: This adds 3x to the left side instead of subtracting it, giving 10x < 12.

Answer: C

Correct answer: x ≥ 10. Multiply every term by the least common denominator, 10:

Subtract 2x and add 10:

Divide by 3:

• A: The original sign is ≥, so 10 itself is a solution and the sign must stay ≥.
• B: This multiplies only the fractions by 10 and leaves the constants 1 and 2 unchanged.
• C (correct): Multiplying every term by 10 gives 5x − 10 ≥ 2x + 20, so 3x ≥ 30 and x ≥ 10.
• D: Multiplying by the positive number 10 never reverses the sign.

Answer: C

Correct answer: [-3, 5). Add 1 to all three parts:

Divide all parts by 2:

The included left end gets a bracket and the excluded right end gets a parenthesis: $[-3,5)$.

• A: The endpoint symbols are reversed: −3 is included (≤) and 5 is not (<).
• B: This subtracts 1 from each part instead of adding 1.
• C (correct): Adding 1 to all parts and dividing by 2 gives −3 ≤ x < 5, with −3 included and 5 excluded.
• D: The right part of the inequality is strict, so 5 needs a parenthesis.

Answer: A

Correct answer:

The expression means x + 3 is within 5 units of zero:

Subtract 3 from all parts:

• A (correct): Writing −5 < x + 3 < 5 and subtracting 3 from every part gives −8 < x < 2.
• B: The original sign is strict, so the endpoints −8 and 2 are not included.
• C: This adds 3 instead of subtracting it; the center of the interval is −3, not 3.
• D: A less-than absolute value means a small distance, which gives a single between-statement, not two pieces.

Answer: B

Correct answer: 18 boxes. Set up

Subtract 190:

so b is at most about 18.4. Boxes are whole, so round down to 18; check:

which is allowed.

• A: Rounding 18.4 up breaks the limit: 190 + 55(19) = 1,235 lb, which exceeds 1,200 lb.
• B (correct): Solving 190 + 55b ≤ 1,200 gives b ≤ about 18.4, and rounding down keeps the load within the limit.
• C: This divides 1,200 by 55 and ignores the driver's weight.
• D: This ignores the driver and also rounds 21.8 up, so the van would be overloaded.

Answer: A

Correct answer: 103 candles. Set up

Subtract 230:

so c is at least about 102.9. For a minimum, round up to 103; check:

which meets the goal.

• A (correct): Solving 230 + 7c ≥ 950 gives c ≥ about 102.9, and a minimum count must be rounded up.
• B: This adds the $230 to the goal instead of subtracting it.
• C: Rounding down a minimum falls short: 230 + 714 = $944, which is less than $950.
• D: This divides the full $950 by 7 and ignores the $230 already raised.

Answer: D

Correct answer:

or x ≥ 4. Split into two pieces:

The first gives

The second gives

• A: The sign is ≥, so the endpoints −3 and 4 belong in the solution.
• B: This subtracts 1 instead of adding 1 when isolating 2x in each piece.
• C: A greater-than absolute value describes values far from the center, so the answer is two pieces joined by or, not the middle segment.
• D (correct): Splitting into 2x − 1 ≥ 7 or 2x − 1 ≤ −7 gives x ≥ 4 or x ≤ −3, with both endpoints included.

Answer: B

Correct answer:

Distribute:

Subtract 4x from both sides:

That statement is false no matter what x is, so the inequality has no solution.

• A: The x terms cancel and leave 8 > 1, which is always true, so every real number is a solution.
• B (correct): Distributing gives 4x − 8 > 4x + 1; the x terms cancel, leaving −8 > 1, which is always false.
• C: The x terms do not cancel; solving gives 2x > 9, so x > 4.5 is the solution set.
• D: The x terms do not cancel; solving gives −x > 7, so x < −7 is the solution set.

Answer: D

Correct answer: x < 6. Multiply every term by 10:

Add 4x and 27 to both sides:

Divide by 7:

which is

Check x = 0:

is true.

• A: The original sign is strict, so 6 itself is not a solution.
• B: This multiplies only the x terms by 10 and leaves 1.5 and 2.7 unchanged.
• C: If you collect x on the left you get −7x > −42; dividing by −7 requires a flip, which was skipped.
• D (correct): Clearing decimals gives 15 − 4x > 3x − 27, so 42 > 7x, which means x < 6.

Answer: D

Correct answer: 8. Subtract 1 from all parts:

Divide by 3:

The integers are −3, −2, −1, 0, 1, 2, 3 and 4, so there are 8.

• A: This includes −4, but the left sign is strict, so −4 is not a solution.
• B: This divides only the middle part by 3, giving −12 < x ≤ 12.
• C: This leaves out 4, but the right sign is ≤, so 4 is included.
• D (correct): The solution is −4 < x ≤ 4, so the integers are −3 through 4, which is 8 values.

Answer: B

Correct answer: 7 or 8 vans. Seating:

so v is at least 6.5; round up to 7. Budget:

so v is at most about 8.57; round down to 8. Both conditions hold for 7 or 8 vans.

• A: Eight vans also work: they seat 64 people and cost $560, within the budget.
• B (correct): Seating requires v ≥ 6.5, so at least 7; the budget requires v ≤ about 8.57, so at most 8.
• C: Nine vans cost $630, over the budget; a maximum count must be rounded down.
• D: Six vans seat only 48 people; a minimum count must be rounded up, not down.

Answer: A

Correct answer: (-∞, -3) ∪ [4, ∞). First part:

divide by −2 and flip to get

Second part:

Or means combine the sets: $(-\infty ,-3)\cup [4,\infty )$.

• A (correct): The first part gives x < −3 (open at −3) and the second gives x ≥ 4 (closed at 4); or joins them with a union.
• B: This forgets to flip when dividing by −2, giving x > −3, which then swallows the second piece.
• C: The first inequality is strict, so −3 is not included and needs a parenthesis.
• D: Infinity always takes a parenthesis, never a bracket.

Answer: C

Correct answer:

Subtract 7 from both sides:

Divide both sides by −3 and reverse the sign:

Check

which is greater than 22.

• A: This divides by −3 but forgets to reverse the inequality sign.
• B: This reverses the sign but drops the negative when dividing 15 by −3.
• C (correct): Subtracting 7 gives −3x > 15; dividing by −3 reverses the sign, giving x < −5.
• D: This adds 7 to 22 instead of subtracting it before dividing by −3.

Answer: A

Correct answer: [−4, 7). The overlap of the two conditions is

List the left endpoint, then the right. Use a bracket for an included endpoint (≤ or ≥) and a parenthesis for an excluded one (< or >): [−4, 7).

• A (correct): −4 is included, so it gets a square bracket; 7 is not included, so it gets a parenthesis.
• B: This swaps the endpoint symbols: it leaves out −4 and includes 7.
• C: This leaves out −4, but the at-least sign means −4 is a solution.
• D: This includes 7, but the strict less-than sign means 7 is not a solution.

Answer: C

Correct answer: x ≥ 10. 0.06 has two decimal places, so multiply every term by 100:

Subtract 6x and add 90: 240 ≤ 24x. Divide by 24: 10 ≤ x, which is x ≥ 10. The multiplier and divisor are positive, so the sign never flips.

• A: This reverses the sign, but the only division is by positive 24, so no reversal is needed.
• B: This subtracts 90 from 150 instead of adding 90 to both sides, giving 60 ≤ 24x.
• C (correct): Multiplying by 100 gives 6x + 150 ≤ 30x − 90, so 240 ≤ 24x and x ≥ 10.
• D: This multiplies the x terms by 100 but the constants only by 10, giving 6x + 15 ≤ 30x − 9.

Answer: A

Correct answer:

or x > 3. A greater-than absolute value means the expression is far from 0, so split it with or:

The first gives 3x > 9, so x > 3. The second gives

Check

|−13| = 13 > 11.

• A (correct): Splitting gives 3x + 2 > 11 or 3x + 2 < −11, so x > 3 or x < −13/3.
• B: This is the solution of |3x + 2| < 11; a greater-than absolute value splits into two outer pieces joined by or.
• C: This handles the negative piece wrongly: 3x + 2 < −11 gives 3x < −13, not 3x < −9.
• D: This assumes the answer is symmetric about 0, but the +2 shifts the center to −2/3.

Answer: D

Correct answer: n ≥ 18. No fewer than, at least and a minimum of all mean greater than or equal to. The number on duty can be 18 or any larger number, so n ≥ 18.

• A: This leaves out exactly 18 crew members, which the rule allows.
• B: This sets a maximum of 18, but the rule sets a minimum.
• C: This describes fewer than 18, the opposite of what the rule requires.
• D (correct): No fewer than 18 means 18 or more, so 18 itself is allowed.

Answer: D

Correct answer: 201 miles. Write B < A as

Subtract 0.10m and 40: 30 < 0.15m. Divide by 0.15: m > 200. Because the sign is strict, 200 itself does not work (both cost $90), so the smallest whole number is 201.

• A: At exactly 200 miles both rentals cost $90, so B is not yet cheaper; the inequality is strict.
• B: At 199 miles B costs $89.90 and A costs $89.75, so A is still cheaper.
• C: This divides the $30 difference in fees by $0.25 instead of by the $0.15 difference in mileage rates.
• D (correct): 70 + 0.10m < 40 + 0.25m gives 30 < 0.15m, so m > 200; the smallest whole number above 200 is 201.

Answer: B

Correct answer: x > 15. The LCD of 3 and 5 is 15. Multiply every term by 15:

Subtract 3x:

Add 30 and divide by 2: x > 15. Check x = 30:

which is greater than 6.

• A: This reverses the sign, but multiplying by 15 and dividing by 2 are both positive steps.
• B (correct): Multiplying by 15 gives 5x − 30 > 3x, so 2x > 30 and x > 15.
• C: This multiplies the fractions by 15 but not the constant 2, giving 5x − 2 > 3x.
• D: This is a sign slip when moving −30 to the other side; it should become +30.

Answer: C

Correct answer: An open circle at 9 with shading to the left. A strict sign (< or >) uses an open circle because the boundary is not a solution. Values less than 9 lie to the left of 9, so shade left. Test 0: 0 < 9 is true, and 0 is on the left.

• A: A closed circle would include 9, which is used for at most, not for strictly less than.
• B: Shading right shows numbers greater than 9, which do not satisfy x < 9.
• C (correct): The strict sign excludes 9, so the circle is open, and smaller numbers lie to the left.
• D: This shows x ≥ 9: it includes 9 and points the wrong way.

Answer: C

Correct answer:

Do each step to all three parts. Subtract 5:

Divide by −2 and flip both signs:

Written from smallest to largest, that is

Check x = 6:

which is allowed by ≤.

• A: The endpoints are right, but the included and excluded ends are swapped after flipping.
• B: This divides by 2 instead of −2, which changes the signs of the endpoints and skips the flip.
• C (correct): Subtracting 5 gives −12 ≤ −2x < 4; dividing by −2 flips both signs, giving 6 ≥ x > −2.
• D: This describes the region outside the endpoints; a three-part inequality is an and statement.

Answer: B

Correct answer: x ≥ 5. Distribute on both sides:

Subtract 5x and add 2: 3x ≥ 15. Divide by 3: x ≥ 5. Check x = 5: 38 ≥ 38 is true, so 5 is included.

• A: This distributes to the first term only, writing 8x − 1 and 5x + 2.
• B (correct): Distributing gives 8x − 2 ≥ 5x + 13, so 3x ≥ 15 and x ≥ 5.
• C: This reverses the sign, but dividing by positive 3 does not change direction.
• D: This forgets to move the −2 on the left, solving 3x ≥ 13.

Answer: A

Correct answer: A closed circle at −1 shaded left and an open circle at 4 shaded right. An or statement includes every number that satisfies at least one piece, so graph both rays.

gets a closed circle at −1 with shading left. x > 4 gets an open circle at 4 with shading right. The gap between −1 and 4 stays unshaded.

• A (correct): −1 is included (≤) and 4 is not (>), and an or statement keeps both outer pieces.
• B: This swaps the circles: −1 should be included and 4 should not.
• C: Shading between treats the statement as an and overlap; the or solution lies outside.
• D: These rays point toward each other and show the middle values, which satisfy neither condition.

Answer: C

Correct answer:

Solve each part.

gives 3x ≤ 15, so x ≤ 5. An and statement keeps only the overlap:

• A: The endpoints are right, but the inclusion is swapped: −3 is excluded (>) and 5 is included (≤).
• B: This adds 1 to −5 instead of subtracting it, getting 2x > −4.
• C (correct): The first gives x > −3 and the second gives x ≤ 5; and means the overlap.
• D: This subtracts 4 from 11 instead of adding 4, getting 3x ≤ 7.

Answer: D

Correct answer:

Divide both sides by −2 and reverse the sign:

Read from the x side, this is

Check x = 0: 8 ≥ 0 is true, and

• A: This divides by −2 but keeps the original direction instead of reversing it.
• B: This flips the sign but drops the negative when dividing 8 by −2.
• C: This divides by 2 instead of −2, losing both the negative and the needed flip.
• D (correct): Dividing both sides by −2 flips the sign: −4 ≤ x, which reads x ≥ −4.

Answer: B

Correct answer: All real numbers. Distribute:

Subtract 6x from both sides:

The variable is gone and the remaining statement is always true, so every real number is a solution.

• A: This would require a false statement after the x terms cancel, but −8 ≤ 1 is true.
• B (correct): Distributing gives 6x − 8 ≤ 6x + 1; the x terms cancel and −8 ≤ 1 is always true.
• C: Once the x terms cancel there is no boundary; every value of x works, including positive ones.
• D: This multiplies 2 by 3x to get 5x instead of 6x, so the x terms fail to cancel and a false boundary appears.

Answer: D

Correct answer:

Divide both sides by 4. Since 4 is positive, the sign stays the same:

A negative number on the right side does not cause a flip.

• A: This flips the sign because the right side is negative, but only dividing by a negative number reverses it.
• B: This drops the negative sign of −28 when dividing.
• C: This subtracts 4 instead of dividing by 4; 4x means 4 times x.
• D (correct): Dividing by positive 4 keeps the direction, so x < −7.

Answer: A

Correct answer: At 4, with a closed circle. Subtract 2x and add 3: 3x ≤ 12, so x ≤ 4. Check the boundary: at x = 4 both sides equal 17, so 4 is a solution and gets a closed circle. Shade to the left.

• A (correct): Solving gives 3x ≤ 12, so x ≤ 4; the ≤ sign includes 4, so the circle is filled.
• B: The boundary is right, but an open circle is only for strict signs like < and >.
• C: This subtracts 3 from 9 instead of adding 3 to both sides, giving 3x ≤ 6.
• D: This is a sign slip; at x = −4 the two sides are −23 and 1, which are not equal.

Answer: D

Correct answer: 42.75 mm. |L − 42.5| ≤ 0.2 means

Add 42.5 to all parts:

Both endpoints are allowed because the sign is ≤. Only 42.75 mm lies outside this range.

• A: |42.3 − 42.5| = 0.2, which meets the at-most condition; the boundary is included.
• B: |42.7 − 42.5| = 0.2, which is exactly the limit, and ≤ includes the boundary.
• C: |42.45 − 42.5| = 0.05, well within the 0.2 mm allowance.
• D (correct): |42.75 − 42.5| = 0.25, which is more than 0.2, so this part is outside the allowed range.

Answer: B

Correct answer: Dividing both sides by −5. Only multiplying or dividing both sides by a negative number reverses the sign. For example, 2 < 5, but dividing by −5 gives

• A: Adding or subtracting any number, positive or negative, never changes the direction.
• B (correct): Dividing by a negative number reverses the order of numbers, so the sign must flip.
• C: Adding a negative number is still addition, which keeps the direction.
• D: Dividing by a positive number keeps the direction of the sign.

Answer: A

Correct answer: 11 tickets. Total spending cannot exceed $250:

Combine:

so 18t ≤ 198 and t ≤ 11. Check:

which is allowed, while 12 tickets would cost $268.

• A (correct): 32 + 20 + 18t ≤ 250 gives 18t ≤ 198, so t ≤ 11; exactly 11 tickets uses the full $250.
• B: This treats the limit as strict; spending exactly $250 is allowed, so 11 tickets works.
• C: This forgets the $20 food money, giving 18t ≤ 218 and t ≤ 12.1.
• D: This ignores both the parking fee and the food money, dividing $250 by $18.

Answer: D

Correct answer:

Undo division by 4 by multiplying both sides by 4. Since 4 is positive, the sign stays:

Check

is true.

• A: This reverses the sign, but multiplying by positive 4 does not change direction.
• B: This divides −3 by 4 instead of multiplying, which does not undo the division.
• C: This subtracts 4 from −3 instead of multiplying both sides by 4.
• D (correct): Multiplying both sides by positive 4 keeps the direction: x ≥ −12.

Answer: B

Correct answer: 68 bags. Write the minimum condition:

Subtract 640: 35b ≥ 2,360. Divide by 35: b ≥ 67.4 (approximately). Bags are whole, and for a minimum you round up, so b = 68. Check:

meets the goal, but 67 bags give 2,985 lb.

• A: This rounds down; 67 bags give only 2,985 lb, which is short of 3,000.
• B (correct): 640 + 35b ≥ 3,000 gives b ≥ 67.4..., so round up to 68; 68 bags bring the total to 3,020 lb.
• C: This ignores the 640 lb already in place and divides 3,000 by 35.
• D: This adds 640 to 3,000 instead of subtracting it before dividing by 35.

Answer: C

Correct answer:

Multiply both sides by the LCD, 6, a positive number, so the sign stays:

Distribute:

Add 2x and subtract 12:

Divide by 5:

Check

(2 + 2)/3 ≈ 1.33 and

and 1.33 ≥ 1 is true.

• A: This reverses the sign, but the only division is by positive 5 after moving the x terms right.
• B: This writes 2(2 − x) as 4 + 2x, losing the negative sign on x.
• C (correct): Multiplying by 6 gives 2(2 − x) ≥ 3(x + 4), so 4 − 2x ≥ 3x + 12, −8 ≥ 5x and x ≤ −8/5.
• D: This multiplies each numerator by its own denominator, writing 3(2 − x) ≥ 2(x + 4).

Answer: B

Correct answer:

On a number line, a number to the right is greater. Although 5 is greater than 2, −2 is to the right of −5, so

This is why multiplying by a negative reverses an inequality.

• A: −7 lies to the left of −3, so it is smaller, not greater.
• B (correct): −2 lies to the right of −5 on the number line, so it is the greater number.
• C: −1 lies to the right of −4, so −1 is greater, not less.
• D: 0 lies to the right of every negative number, so 0 is greater than −6.

Answer: A

Correct answer: x > 5. Add 4x to both sides to keep a positive coefficient:

Add 6: 15 < 3x. Divide by 3: 5 < x, which is x > 5. Check x = 6:

which is less than −12.

• A (correct): Adding 4x gives 9 < 3x − 6, so 15 < 3x and x > 5.
• B: This reverses the sign without dividing by a negative; the coefficient 3 is positive.
• C: This comes from a sign slip that produces −15 < 3x instead of 15 < 3x.
• D: This subtracts 6 from 9 instead of adding 6 to both sides, getting 3 < 3x.