Linear Inequalities - SAT Math

On a number line, the solution set is shown with open circles at $-1$ and $4$ and shading between them only. Which inequality matches the graph?
Answer: $-1<x<4$ This problem asks us to identify which inequality matches a graph showing open circles at -1 and 4 with shading between them. Open circles indicate that the endpoints are NOT included in the solution set, meaning we use strict inequalities (< or >) rather than ≤ or ≥. Since the shading is between -1 and 4, the solution includes all values greater than -1 AND less than 4, which is written as $-1 < x < 4$ . A common mistake is using closed inequalities (≤ or ≥) when you see open circles, or writing the solution as two separate parts instead of a compound inequality. When interpreting number line graphs, remember: open circle means strict inequality, closed circle means inclusive inequality.
A science club sells candles for $7 each to raise money. The club needs to raise at least $210 but no more than $350. If $x$ is the number of candles sold, which compound inequality represents the possible values of $x$ ?
Answer: $210\le 7x\le 350$ The club needs to raise between $210 and $350 by selling candles at $7 each. If x is the number of candles sold, then the revenue is 7x dollars. We need 7x to be at least 210 but no more than 350, which gives us the compound inequality 210 ≤ 7x ≤ 350. This represents that the revenue must be greater than or equal to $210 AND less than or equal to $350. Common errors include using the wrong inequality symbols or writing the compound inequality in the wrong order. When setting up compound inequalities for range problems, ensure the smaller value is on the left.
Solve the inequality and choose the correct solution set: $\frac{x-4}{3}+2>5$
Answer: $x>13$ We need to solve $(x-4)/3 + 2 > 5$ step by step. First, subtract 2 from both sides to get $(x-4)/3 > 3$ , then multiply both sides by 3 to get $x - 4 > 9$ . Finally, add 4 to both sides to get $x > 13$ . The key insight is that we're multiplying by positive 3, so the inequality sign stays the same direction. A common error would be to incorrectly simplify the left side or to include equality when the original inequality is strict (>). When solving inequalities with fractions, clear the denominator early by multiplying both sides by the positive denominator.
A phone plan costs $40 plus $0.10x$ dollars for $x$ text messages. If the total cost is less than $55, what is the greatest whole number of texts $x$ allowed?
Answer: $149$ The total cost is $40 + $0.10x, and this must be less than $55, giving us 40 + 0.10x < 55. Subtracting 40 from both sides: 0.10x < 15. Dividing by 0.10 (or multiplying by 10): x < 150. Since x must be a whole number of texts, the greatest value is x = 149. The key is recognizing that "less than" (not "less than or equal to") means we cannot include 150. When finding the "greatest whole number" with a strict inequality, you must round down.
Solve the compound inequality $-1\le 2x+3<9$ . What is the solution written in interval form?
Answer: $[-2,3)$ We need to solve the compound inequality -1 ≤ 2x + 3 < 9. This breaks into two parts: -1 ≤ 2x + 3 AND 2x + 3 < 9. For the first part: subtract 3 to get -4 ≤ 2x, then divide by 2 to get -2 ≤ x. For the second part: subtract 3 to get 2x < 6, then divide by 2 to get x < 3. Combining these gives -2 ≤ x < 3, which in interval notation is [-2, 3). The square bracket at -2 means it's included (closed interval), while the parenthesis at 3 means it's excluded (open interval).
Solve the compound inequality $-2\le 3x+1<10$ . Which interval describes the solution set?
Answer: $-1\le x<3$ The question requires solving -2 ≤ 3x + 1 < 10 and identifying the interval. Subtract 1 from all parts: -3 ≤ 3x < 9. Divide by 3 (positive, so no reversal): -1 ≤ x < 3. This includes x = -1 but excludes x = 3. A common error is mishandling the compound structure, like reversing unnecessarily. Another pitfall is adding instead of subtracting, shifting bounds wrongly. When solving compounds, apply operations to all parts and verify with test points like x = -1 and x = 2.9.
A theater must sell at least 180 tickets to break even. They have already sold 47 tickets and will sell $x$ more. Which inequality represents the situation?
Answer: $47 + x \ge 180$ The theater needs to sell at least 180 tickets total to break even. They've already sold 47 tickets and will sell x more, so the total tickets sold is 47 + x. Since this must be at least 180, the inequality is 47 + x ≥ 180. A common error is to multiply instead of add when combining quantities - remember that "x more" means addition, not multiplication. The phrase "at least" translates to the ≥ symbol.
Solve the compound inequality $-1\le 2x+5<9$ . Which interval is the solution?
Answer: $-3\le x<2$ This question asks to solve the compound inequality -1 ≤ 2x + 5 < 9 and find the solution interval. Subtract 5 from all parts: -6 ≤ 2x < 4. Divide by 2: -3 ≤ x < 2. A key error is forgetting to apply operations to all parts of the compound inequality, which can distort the solution. Another common mistake is reversing inequalities unnecessarily, but here division by positive 2 keeps the directions the same. To verify, test boundary points like x = -3 and x = 2 in the original inequality.
A water tank contains 18 liters and is being drained at 1.5 liters per minute. If at least 6 liters must remain, after how many minutes $t$ is the inequality satisfied?
Answer: $t \le 8$ The tank starts with 18 liters and loses 1.5 liters per minute, so after t minutes it has 18 - 1.5t liters. We need at least 6 liters to remain, giving us 18 - 1.5t ≥ 6. Subtracting 18 from both sides: -1.5t ≥ -12. Dividing by -1.5 (and reversing the inequality): t ≤ 8. This means the draining can continue for at most 8 minutes. The common error is forgetting to reverse the inequality when dividing by negative 1.5. Check: after 8 minutes, 18 - 1.5(8) = 18 - 12 = 6 liters remain.
A number line shows a closed circle at $4$ with shading to the left. Which inequality is shown?
Answer: $x\le 4$ The question asks which inequality matches a number line with a closed circle at 4 and shading to the left. Closed circle means inclusive (≤ or ≥), and left shading indicates less than. Thus, it's x ≤ 4. A common error is misreading the shading direction as greater than. Another pitfall is confusing closed with open, leading to strict inequality. When interpreting graphs, remember left shading is less, right is greater, and circle type determines equality.
Solve the inequality $4x-9 \le 3x+2$ .
Answer: $x \le 11$ To solve 4x - 9 ≤ 3x + 2, we need to isolate x. Subtracting 3x from both sides gives x - 9 ≤ 2. Adding 9 to both sides yields x ≤ 11. The key technique is to move all x terms to one side and constants to the other. A common error is sign mistakes when moving terms - remember that subtracting 3x from both sides is different from adding -3x. Always verify by substituting: if x = 11, then 4(11) - 9 = 35 and 3(11) + 2 = 35, so 35 ≤ 35 ✓.
A store is choosing the number of notebooks $n$ to order. The supplier requires either fewer than 20 notebooks or at least 50 notebooks (no orders from 20 to 49). Which inequality represents this restriction?
Answer: $n<20\text{ or }n\ge 50$ This problem describes a situation where notebook orders must be either "fewer than 20" OR "at least 50," with no orders allowed between 20 and 49. The phrase "fewer than 20" translates to $n < 20$ , and "at least 50" translates to $n ≥ 50$ . Since the supplier accepts orders in either range (not both simultaneously), we connect these with "or," giving us $n < 20$ or $n ≥ 50$ . This creates two separate regions on the number line with a gap from 20 to 49. A common mistake is using "and" instead of "or," but $n < 20$ and $n ≥ 50$ is impossible since no number can be both less than 20 and at least 50 simultaneously. When you see restrictions that create separate acceptable ranges, use "or" to connect the inequalities.
Solve $\dfrac{x-3}{4}\ge -2$ . Which statement gives the correct solution set?
Answer: $x\ge -5$ The question asks to solve (x - 3)/4 ≥ -2 and select the solution set. Multiply both sides by 4 (positive, no flip): x - 3 ≥ -8. Add 3: x ≥ -5. A common error is flipping when multiplying by positive 4. Another pitfall is subtracting instead of adding 3. For fractional inequalities, clear the denominator first, then isolate; test x=-5 in original to verify.
For a lab mixture, the pH level $x$ must be between 6 and 8 inclusive. Which compound inequality represents the allowed values of $x$ ?
Answer: $6\le x\le 8$ The problem states that pH level x must be between 6 and 8 inclusive, meaning x can equal 6 or 8. This translates to the compound inequality 6 ≤ x ≤ 8, which means x is greater than or equal to 6 AND less than or equal to 8. The word "inclusive" is crucial here - it means the endpoints are included, so we use ≤ rather than <. A common mistake is using strict inequalities (< or >) when the problem says "inclusive" or "between...and...inclusive." When you see "inclusive," always use ≤ or ≥ to include the boundary values.
A gym charges a $25 signup fee plus $x$ dollars per month. If you can spend at most $85 total for the first month, for what values of $x$ is the inequality true?
Answer: $x \le 60$ This problem asks for the monthly cost x that keeps the total first-month expense at most $85. The total cost is the $25 signup fee plus x dollars for the monthly charge, which must be less than or equal to $85, giving us 25 + x ≤ 85. Subtracting 25 from both sides yields x ≤ 60. The key error to avoid is confusing "at most" with "at least" - "at most $85" means we can spend $85 or less, not more. When you see "at most," think of the ≤ symbol.
A gym charges a $25 sign-up fee plus $8 per class. You have at most $97 to spend. Let $c$ be the number of classes you can take. Which inequality represents this situation, and what whole-number values of $c$ satisfy it?
Answer: $25+8c\le97;\ c\le9$ This question asks for the inequality modeling a gym membership with a $25 sign-up fee and $8 per class, where you can spend at most $97, and the whole-number values of c that satisfy it. The total cost is 25 + 8c, which must be less than or equal to 97, so the inequality is 25 + 8c ≤ 97. Subtract 25 from both sides to get 8c ≤ 72, then divide by 8 to find c ≤ 9. Since c is a whole number, possible values are c = 0 to 9. A common error is using a greater-than inequality, which would incorrectly suggest spending at least $97. Another pitfall is excluding c = 9, but at c = 9 the cost is exactly $97, which is allowed under 'at most.' When modeling real-world constraints, always check boundary values to ensure the inequality direction matches the phrase like 'at most' or 'at least.'
Solve $7-2x>19$ . What is the solution for $x$ ?
Answer: $x<-6$ We need to solve 7 - 2x > 19. First, subtract 7 from both sides: -2x > 12. Now divide both sides by -2, and since we're dividing by a negative number, we MUST reverse the inequality sign: x < -6. The critical step is remembering to flip the inequality when dividing by -2. This is one of the most common errors in solving linear inequalities - always reverse the inequality sign when multiplying or dividing by a negative number.
A number line shows a closed circle at $1$ with shading to the left. Which inequality is represented by this graph?
Answer: $x\le1$ This question requires identifying the inequality for a number line with a closed circle at 1 and shading to the left. The closed circle means x = 1 is included, and shading left indicates all values less than 1, so x ≤ 1. A common pitfall is confusing left shading with greater-than inequalities; remember left means less than. Another error is using an open circle, which would make it strict inequality. When interpreting graphs, note the circle type and shading direction carefully to match the inequality symbol.
A delivery driver's route length $x$ (in miles) must be more than 12 miles but no more than 30 miles. Which compound inequality describes all possible values of $x$ ?
Answer: $12<x\le 30$ This question asks for a compound inequality representing route lengths that are "more than 12 miles but no more than 30 miles." The phrase "more than 12" translates to $x > 12$ (strict inequality, not including 12). The phrase "no more than 30" means "at most 30," which translates to $x ≤ 30$ (including 30). Combining these with "and" gives us $12 < x ≤ 30$ . The key distinction is recognizing when to use strict (<, >) versus non-strict (≤, ≥) inequalities based on the wording. Common errors include using the same type of inequality for both bounds or confusing "and" with "or"—remember that "but" in this context means "and," creating a range between the values.
A delivery truck can carry at most 1,200 lb. The empty truck weighs 3,800 lb, and the loaded weight is $3,800 + 50x$ where $x$ is the number of identical boxes (50 lb each). What is the greatest integer value of $x$ that satisfies the constraint?
Answer: $24$ The truck can carry at most 1,200 lb, and the loaded weight is 3,800 + 50x. Setting up the inequality: 3,800 + 50x ≤ 5,000 (since 3,800 + 1,200 = 5,000 total capacity). Solving: 50x ≤ 1,200, so x ≤ 24. Since x must be a whole number of boxes, the greatest integer value is x = 24. Always check: 3,800 + 50(24) = 3,800 + 1,200 = 5,000 lb, which equals the maximum capacity. When asked for the "greatest integer value," round down from your decimal answer.

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