Solving Linear Inequalities - Math 1

1. Which number line correctly represents the solution to $\frac{5-3x}{2} \geq x + 1$?
   Answer: Closed circle at $\frac{3}{5}$, arrow pointing left Starting with $\frac{5-3x}{2} \geq x + 1$, multiply both sides by 2: $5-3x \geq 2x + 2$. Subtract $2x$ and subtract 5: $-3x - 2x \geq 2 - 5$, so $-5x \geq -3$. Dividing by $-5$ (and flipping the inequality): $x \leq \frac{3}{5}$. This requires a closed circle at $\frac{3}{5}$ (since equality is included) with an arrow pointing left. Choice A has the wrong boundary point. Choice B has the wrong boundary point. Choice C has the correct boundary but wrong direction and wrong circle type.
2. A rental car company charges a base fee of $30 plus $0.25 per mile driven. If Maria's budget allows her to spend at most $85 on the rental, what is the maximum number of complete miles she can drive?
   Answer: 220 miles Let $m$ be the number of miles. The total cost is $30 + 0.25m \leq 85$. Subtracting 30: $0.25m \leq 55$. Dividing by 0.25: $m \leq 220$. Since we need complete miles, the maximum is exactly 220 miles. Choice A (219) is one mile short of the maximum. Choice C (221) exceeds the budget by $0.25. Choice D results from incorrectly dividing 85 by 0.25 without subtracting the base fee.
3. Which inequality represents the possible number of miles $m$ the customer can travel?
   Answer: $\frac{50}{3} \leq m \leq 30$ The total cost is $2.50 + 0.75m$. The constraint is $15 \leq 2.50 + 0.75m \leq 25$. Subtracting 2.50: $12.50 \leq 0.75m \leq 22.50$. Dividing by 0.75: $\frac{12.50}{0.75} \leq m \leq \frac{22.50}{0.75}$. Converting to fractions: $\frac{12.5}{0.75} = \frac{125}{7.5} = \frac{1250}{75} = \frac{50}{3}$ and $\frac{22.5}{0.75} = \frac{225}{7.5} = \frac{2250}{75} = 30$. Choice B gives the decimal approximation of $\frac{50}{3} \approx 16.67$. Choice C uses incorrect fraction arithmetic. Choice D uses the values before dividing by 0.75.
4. A student claims that the solution to $\frac{2x-1}{3} \geq \frac{x+2}{-2}$ is $x \leq -\frac{7}{5}$. Which error did the student most likely make?
   Answer: Did not reverse the inequality when multiplying by a negative number The correct solution: $\frac{2x-1}{3} \geq \frac{x+2}{-2}$ becomes $\frac{2x-1}{3} \geq -\frac{x+2}{2}$. Cross-multiplying: $2(2x-1) \geq 3(-1)(x+2)$, so $4x-2 \geq -3x-6$. This gives $7x \geq -4$, so $x \geq -\frac{4}{7}$. The student got $x \leq -\frac{7}{5}$, suggesting they cross-multiplied correctly but forgot to reverse the inequality sign when multiplying by the negative denominator. The other errors would lead to different incorrect results.
5. A manufacturer's profit $P$ (in thousands of dollars) is given by $P = 15x - 3$, where $x$ is the number of units produced (in hundreds). If the manufacturer wants a profit of at least $18,000 but no more than $42,000, which inequality represents the constraint on production?
   Answer: $1.4 \leq x \leq 3$ First, convert the profit constraints to thousands: $18,000 = 18 thousand and $42,000 = 42 thousand. Set up the compound inequality: $18 \leq 15x - 3 \leq 42$. Adding 3 to all parts: $21 \leq 15x \leq 45$. Dividing by 15: $1.4 \leq x \leq 3$. Choice B incorrectly multiplies by 100 instead of converting units. Choice C uses the wrong profit values (12 and 42 instead of 18 and 42). Choice D makes both unit conversion and calculation errors.
6. The inequality $|2x - 6| \leq 4$ is equivalent to which compound linear inequality?
   Answer: $1 \leq x \leq 5$ The absolute value inequality $|2x - 6| \leq 4$ means $-4 \leq 2x - 6 \leq 4$. Adding 6 to all parts: $2 \leq 2x \leq 10$. Dividing by 2: $1 \leq x \leq 5$. Choice B shows the intermediate step but not the final simplified form. Choice C represents the solution to $|2x - 6| \geq 4$. Choice D shows the compound inequality for $|2x - 6| \geq 4$ before solving.
7. The inequality $|3x - 6| \leq 9$ is equivalent to which compound inequality?
   Answer: $-1 \leq x \leq 5$ The absolute value inequality $|3x - 6| \leq 9$ is equivalent to $-9 \leq 3x - 6 \leq 9$. Adding 6 to all parts: $-3 \leq 3x \leq 15$. Dividing by 3: $-1 \leq x \leq 5$. Choice B forgets to add 6 before dividing. Choice C represents the solution to $|3x - 6| \geq 9$. Choice D forgets to account for the $3x - 6$ inside the absolute value.
8. A student solving $\frac{2x - 4}{-3} \geq 6$ writes the solution as $x \geq -11$. What mistake did the student make, and what is the correct solution?
   Answer: Failed to reverse the inequality; correct solution is $x \leq -7$ Starting with $\frac{2x - 4}{-3} \geq 6$, multiply both sides by $-3$ (must reverse the inequality): $2x - 4 \leq -18$. Adding 4: $2x \leq -14$. Dividing by 2: $x \leq -7$. The student's answer $x \geq -11$ suggests they forgot to reverse the inequality sign when multiplying by the negative number $-3$. The correct solution is $x \leq -7$.
9. A company's profit $P$ (in thousands of dollars) is modeled by $P = 2x - 15$, where $x$ is the number of units sold (in hundreds). For the company to avoid a loss, which interval represents the possible values of $x$?
   Answer: $x \geq 7.5$ To avoid a loss, the profit must be non-negative: $P \geq 0$. Setting up the inequality: $2x - 15 \geq 0$. Adding 15 to both sides: $2x \geq 15$. Dividing by 2: $x \geq 7.5$. Since breaking even (zero profit) still avoids a loss, we use $\geq$ rather than $>$. Choice B uses strict inequality, missing the break-even point. Choices C and D incorrectly solve the inequality by not dividing 15 by 2.
10. A temperature $T$ (in Celsius) is considered safe for a chemical reaction if $2T + 15 \geq 35$ and $3T - 12 \leq 45$. What is the range of safe temperatures?
    Answer: $10° \leq T \leq 19°$ When you encounter compound inequalities like this, you're looking for values that satisfy both conditions simultaneously. This means finding the overlap between the solution sets of each individual inequality. Let's solve each inequality separately. For $2T + 15 \geq 35$, subtract 15 from both sides to get $2T \geq 20$, then divide by 2 to find $T \geq 10$. For $3T - 12 \leq 45$, add 12 to both sides to get $3T \leq 57$, then divide by 3 to find $T \leq 19$. Since the temperature must satisfy both conditions (using "and"), you need the intersection of these ranges. The temperature must be both greater than or equal to 10° AND less than or equal to 19°. This gives us $10° \leq T \leq 19°$. Looking at the wrong answers: Choice A ($T < 10°$ or $T > 19°$) represents the complement of our answer - temperatures that are unsafe. Choice B ($10° < T < 19°$) incorrectly excludes the boundary values 10° and 19°, using strict inequalities instead of the inclusive ones we found. Choice C ($T \leq 10°$ or $T \geq 19°$) also represents unsafe temperatures, mixing up the intersection with union. Remember that "and" in compound inequalities means intersection (overlap), while "or" means union (either condition). Always solve each inequality first, then determine whether you need the values that work for both conditions or either condition.
11. Consider the system of inequalities: $\begin{cases} 2x + y \leq 8 \\ x - y \geq -2 \\ x \geq 0 \end{cases}$ Which point lies within the solution region?
    Answer: $(3, 1)$ Check each point against all three inequalities. For $(3,1)$: $2(3) + 1 = 7 \leq 8$ ✓, $3 - 1 = 2 \geq -2$ ✓, $3 \geq 0$ ✓. All satisfied. For $(4,2)$: $2(4) + 2 = 10 \not\leq 8$ ✗. For $(1,7)$: $2(1) + 7 = 9 \not\leq 8$ ✗. For $(5,-1)$: $2(5) + (-1) = 9 \not\leq 8$ ✗.
12. When solving $\frac{3x - 2}{4} - \frac{x + 1}{3} \geq 2$, what is the correct solution set?
    Answer: $x \geq \frac{34}{5}$ First, find a common denominator of 12: $\frac{3(3x-2)}{12} - \frac{4(x+1)}{12} \geq 2$. This becomes $\frac{9x-6-4x-4}{12} \geq 2$, which simplifies to $\frac{5x-10}{12} \geq 2$. Multiplying both sides by 12: $5x-10 \geq 24$. Adding 10: $5x \geq 34$. Dividing by 5: $x \geq \frac{34}{5}$. Choice B flips the inequality sign incorrectly. Choices C and D result from arithmetic errors in combining fractions.
13. Solve the compound inequality $-3 < 2x + 5 \leq 11$ and express the solution in interval notation.
    Answer: $(-4, 3]$ Working with the compound inequality $-3 < 2x + 5 \leq 11$: First, subtract 5 from all parts: $-8 < 2x \leq 6$. Then divide by 2: $-4 < x \leq 3$. In interval notation, this is $(-4, 3]$. Choice B incorrectly includes -4 and excludes 3. Choice C excludes both endpoints. Choice D incorrectly includes both endpoints.
14. Two inequalities $2x + 3 \geq 7$ and $5 - x > 1$ are solved simultaneously. What values of $x$ satisfy both conditions?
    Answer: $2 \leq x < 4$ When you encounter a system of inequalities, you need to find the values of x that satisfy both conditions simultaneously. This means finding the intersection (overlap) of the solution sets. Let's solve each inequality separately. For $2x + 3 \geq 7$, subtract 3 from both sides: $2x \geq 4$, then divide by 2: $x \geq 2$. For $5 - x > 1$, subtract 5 from both sides: $-x > -4$, then multiply by -1 (which flips the inequality sign): $x < 4$. Now you need values that satisfy both $x \geq 2$ AND $x < 4$. This gives you $2 \leq x < 4$, which is answer choice B. Let's examine why the other options are wrong. Choice A gives $2 < x \leq 4$, which incorrectly excludes $x = 2$ (which does satisfy $2x + 3 \geq 7$) and incorrectly includes $x = 4$ (which violates $5 - x > 1$ since $5 - 4 = 1$, not greater than 1). Choice C represents the union of the regions outside our solution, which would be the opposite of what we want. Choice D also represents values outside our intersection, missing the key word "simultaneously." Remember that "simultaneously" means finding the intersection (AND) of solution sets, not the union (OR). Always solve each inequality separately first, then find where their solutions overlap. Pay careful attention to whether endpoints are included (≤, ≥) or excluded (<, >).

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