Q1: Which of the following is the correct graph of the inequality \(y > 2x - 3\)? (a) A dashed line with shading below the line (b) A solid line with shading above the line (c) A dashed line with shading above the line (d) A solid line with shading below the line
Solution:
Ans: (c) Explanation: The inequality uses the greater than symbol (>), which means the line should be dashed (not included). Since \(y\) is greater than \(2x - 3\), we shade above the line.
Q2: What is the solution to the inequality \(3x - 7 \geq 8\)? (a) \(x \geq 5\) (b) \(x \leq 5\) (c) \(x \geq 1\) (d) \(x \leq 1\)
Solution:
Ans: (a) Explanation: Add 7 to both sides: \(3x \geq 15\). Divide by 3: \(x \geq 5\). The inequality symbol does not flip because we divided by a positive number.
Q3: Which point is a solution to the system of inequalities: \(y < x="" +="" 2\)="" and="" \(y=""> -x + 1\)? (a) (0, 3) (b) (1, 2) (c) (2, 0) (d) (-1, -1)
Solution:
Ans: (b) Explanation: Test point (1, 2): For \(y < x="" +="" 2\):="" \(2="">< 1="" +="" 2\)="" →="" \(2="">< 3\)=""> For \(y > -x + 1\): \(2 > -1 + 1\) → \(2 > 0\) ✓ Both inequalities are satisfied, so (1, 2) is in the solution region.
Q4: When graphing \(y \leq -\frac{1}{2}x + 4\), which type of line should be used? (a) A dashed line (b) A solid line (c) A dotted line (d) No line at all
Solution:
Ans: (b) Explanation: The inequality symbol is less than or equal to (≤), which includes the boundary line. Therefore, a solid line is used to show that points on the line are part of the solution.
Ans: (a) Explanation: Subtract 5 from both sides: \(-4x <> Divide by -4 (remember to flip the inequality symbol): \(x > -3\). When dividing by a negative number, the inequality direction reverses.
Q6: Which system of inequalities has NO solution? (a) \(y > x + 1\) and \(y > x - 2\) (b) \(y < 2x="" +="" 3\)="" and="" \(y=""> 2x + 5\) (c) \(y \geq x\) and \(y \leq -x\) (d) \(y > 0\) and \(x > 0\)
Solution:
Ans: (b) Explanation: The inequalities \(y < 2x="" +="" 3\)="" and="" \(y=""> 2x + 5\) represent parallel regions with no overlap. Since \(2x + 5 > 2x + 3\) for all \(x\), there is no value of \(y\) that can be both less than \(2x + 3\) and greater than \(2x + 5\) simultaneously.
Q7: What is the boundary line equation for the inequality \(2x + 3y \leq 12\)? (a) \(y = -\frac{2}{3}x + 4\) (b) \(y = \frac{2}{3}x + 4\) (c) \(y = -\frac{3}{2}x + 6\) (d) \(y = 2x + 4\)
Solution:
Ans: (a) Explanation: To find the boundary line, replace the inequality with an equals sign: \(2x + 3y = 12\). Solve for \(y\): \(3y = -2x + 12\) → \(y = -\frac{2}{3}x + 4\).
Q8: Which inequality represents all points above the line \(y = 3x - 1\)? (a) \(y < 3x="" -=""> (b) \(y \leq 3x - 1\) (c) \(y > 3x - 1\) (d) \(x > 3y - 1\)
Solution:
Ans: (c) Explanation: Points above a line have \(y\)-values that are greater than the corresponding values on the line. Therefore, the inequality is \(y > 3x - 1\).
Section B: Fill in the Blanks
Q9:When graphing a system of inequalities, the solution is the region where all shaded areas __________.
Solution:
Ans: overlap Explanation: The solution to a system of inequalities is the intersection or overlap of all individual solution regions.
Q10:If you multiply or divide both sides of an inequality by a negative number, you must __________ the inequality symbol.
Solution:
Ans: flip (or reverse) Explanation: This is a fundamental rule of inequalities: multiplying or dividing by a negative number reverses the direction of the inequality symbol.
Q11:The inequality \(y \geq mx + b\) is graphed with a __________ line.
Solution:
Ans: solid Explanation: The symbol ≥ means "greater than or equal to," so the boundary line is included in the solution and must be drawn as a solid line.
Q12:To test which side of the boundary line to shade, substitute a __________ point into the inequality.
Solution:
Ans: test Explanation: A test point (commonly (0,0) if not on the line) is used to determine which side of the boundary satisfies the inequality.
Q13:The solution to the inequality \(x - 5 > 12\) is \(x >\) __________.
Solution:
Ans: 17 Explanation: Add 5 to both sides: \(x > 12 + 5\) → \(x > 17\).
Q14:In a system of inequalities, if the shaded regions do not overlap, the system has __________ solution.
Solution:
Ans: no Explanation: When the shaded regions do not overlap, there are no points that satisfy all inequalities simultaneously, so there is no solution.
Section C: Word Problems
Q15:A student needs to score at least 85 points on a test to get an A. If the student has already earned 62 points, write and solve an inequality to find how many more points \(p\) are needed.
Solution:
Ans: \(62 + p \geq 85\) \(p \geq 23\) Final Answer: The student needs at least 23 more points.
Q16:A cell phone plan costs $30 per month plus $0.10 per text message. If Maya wants to spend no more than $45 per month, write and solve an inequality to find the maximum number of text messages \(t\) she can send.
Solution:
Ans: \(30 + 0.10t \leq 45\) \(0.10t \leq 15\) \(t \leq 150\) Final Answer: Maya can send at most 150 text messages.
Q17:A parking garage charges $5 for the first hour and $3 for each additional hour. If Mr. Chen has $20, write and solve an inequality to find the maximum total number of hours \(h\) he can park.
Solution:
Ans: \(5 + 3(h - 1) \leq 20\) \(5 + 3h - 3 \leq 20\) \(3h + 2 \leq 20\) \(3h \leq 18\) \(h \leq 6\) Final Answer: Mr. Chen can park for at most 6 hours.
Q18:A movie theater sells adult tickets for $12 and student tickets for $8. The theater wants to make at least $240 from ticket sales. Write an inequality that represents the number of adult tickets \(a\) and student tickets \(s\) needed, and determine if selling 10 adult tickets and 15 student tickets meets this goal.
Solution:
Ans: Inequality: \(12a + 8s \geq 240\) Test: \(12(10) + 8(15) = 120 + 120 = 240\) Since \(240 \geq 240\) is true, Final Answer: Yes, selling 10 adult tickets and 15 student tickets meets the goal.
Q19:Graph the system of inequalities: \(y \leq 2x + 1\) and \(y > -x + 3\). Determine if the point (1, 3) is in the solution region.
Solution:
Ans: Test point (1, 3): For \(y \leq 2x + 1\): \(3 \leq 2(1) + 1\) → \(3 \leq 3\) ✓ For \(y > -x + 3\): \(3 > -1 + 3\) → \(3 > 2\) ✓ Final Answer: Yes, the point (1, 3) is in the solution region.
Q20:A farmer has at most 100 acres to plant corn and soybeans. He wants to plant at least 30 acres of corn. Write a system of inequalities for the number of acres of corn \(c\) and soybeans \(s\), and determine if planting 40 acres of corn and 55 acres of soybeans is possible.
Solution:
Ans: System of inequalities: \(c + s \leq 100\) \(c \geq 30\) Test: \(40 + 55 = 95 \leq 100\) ✓ and \(40 \geq 30\) ✓ Final Answer: Yes, planting 40 acres of corn and 55 acres of soybeans is possible.
The document Worksheet (with Solutions): Inequalities (Systems & Graphs) is a part of the Grade 9 Course Integrated Math 1.
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