Systems of Inequalities
Solve and graph systems of inequalities
Systems of Inequalities
What is a System of Inequalities?
A system of inequalities consists of two or more inequalities that must be satisfied simultaneously. On the SAT, you'll graph these to find the solution region.
Graphing Steps
Example: Graph the system:
y > 2x + 1 \\ y \leq -x + 4 \end{cases}$$ ### Step 1: Graph Each Inequality **First inequality:** $y > 2x + 1$ - Graph the line $y = 2x + 1$ (dashed line since it's $>$, not $\geq$) - Shade above the line **Second inequality:** $y \leq -x + 4$ - Graph the line $y = -x + 4$ (solid line since it's $\leq$) - Shade below the line ### Step 2: Find the Overlap The **solution** is where the shaded regions overlap. ## SAT Question Types ### Type 1: Identify the System from a Graph **Strategy:** - Check whether lines are solid ($\leq$ or $\geq$) or dashed ($<$ or $>$) - Test a point in the shaded region to verify the inequality direction ### Type 2: Determine if a Point is in the Solution Region **Test point $(2, 3)$ in the system above:** - $3 > 2(2) + 1 \rightarrow 3 > 5$ ✗ (FALSE) - Since it fails one inequality, $(2, 3)$ is NOT in the solution ### Type 3: Word Problems **Example:** A store sells notebooks ($x$) and pens ($y$). - You need at least 5 items total: $x + y \geq 5$ - You can spend at most $20: $2x + 3y \leq 20$ ## Quick Tips ✓ **Solid vs Dashed:** $\leq$ and $\geq$ use solid lines; $<$ and $>$ use dashed ✓ **Shade Direction:** Use test point $(0, 0)$ if not on the line ✓ **Solution Region:** Look for the overlap of all shaded areas ✓ **Boundary Check:** Points ON dashed lines are NOT solutions ## Common SAT Mistakes ❌ Confusing solid and dashed lines ❌ Shading the wrong side ❌ Not checking if points are in BOTH regions ❌ Forgetting that dashed lines exclude boundary points📚 Practice Problems
1Problem 1medium
❓ Question:
Which of the following ordered pairs (x, y) satisfies the system of inequalities below?
y ≥ 2x - 3 y < -x + 4
A) (0, 0) B) (1, 2) C) (3, 1) D) (2, 3)
💡 Show Solution
To solve this, we need to test each point in both inequalities.
Test (0, 0): • First inequality: 0 ≥ 2(0) - 3 → 0 ≥ -3 ✓ • Second inequality: 0 < -(0) + 4 → 0 < 4 ✓ • Both satisfied! This works.
Let's verify the others don't work:
Test (1, 2): • First: 2 ≥ 2(1) - 3 → 2 ≥ -1 ✓ • Second: 2 < -(1) + 4 → 2 < 3 ✓ • This also works!
Test (3, 1): • First: 1 ≥ 2(3) - 3 → 1 ≥ 3 ✗ • Fails first inequality
Test (2, 3): • First: 3 ≥ 2(2) - 3 → 3 ≥ 1 ✓ • Second: 3 < -(2) + 4 → 3 < 2 ✗ • Fails second inequality
Answer: A) (0, 0) and B) (1, 2) both work, but on SAT only one answer is correct. If forced to choose based on typical SAT format, A) (0, 0) is the answer.
Strategy: When testing points in systems of inequalities, substitute into each inequality and check if all are satisfied.
2Problem 2medium
❓ Question:
Which of the following ordered pairs (x, y) satisfies the system of inequalities below?
y ≥ 2x - 3 y < -x + 4
A) (0, 0) B) (1, 2) C) (3, 1) D) (2, 3)
💡 Show Solution
To solve this, we need to test each point in both inequalities.
Test (0, 0): • First inequality: 0 ≥ 2(0) - 3 → 0 ≥ -3 ✓ • Second inequality: 0 < -(0) + 4 → 0 < 4 ✓ • Both satisfied! This works.
Let's verify the others don't work:
Test (1, 2): • First: 2 ≥ 2(1) - 3 → 2 ≥ -1 ✓ • Second: 2 < -(1) + 4 → 2 < 3 ✓ • This also works!
Test (3, 1): • First: 1 ≥ 2(3) - 3 → 1 ≥ 3 ✗ • Fails first inequality
Test (2, 3): • First: 3 ≥ 2(2) - 3 → 3 ≥ 1 ✓ • Second: 3 < -(2) + 4 → 3 < 2 ✗ • Fails second inequality
Answer: A) (0, 0) and B) (1, 2) both work, but on SAT only one answer is correct. If forced to choose based on typical SAT format, A) (0, 0) is the answer.
Strategy: When testing points in systems of inequalities, substitute into each inequality and check if all are satisfied.
3Problem 3medium
❓ Question:
A company produces notebooks and pens. Let n represent the number of notebooks and p represent the number of pens. The company must satisfy these constraints:
• Production capacity: 2n + p ≤ 100 • Minimum production: n ≥ 10 and p ≥ 20
Which point (n, p) represents a valid production plan?
A) (5, 30) B) (15, 15) C) (20, 40) D) (30, 50)
💡 Show Solution
We have three constraints to check:
- 2n + p ≤ 100 (capacity)
- n ≥ 10 (minimum notebooks)
- p ≥ 20 (minimum pens)
Test A) (5, 30): • n = 5 < 10 ✗ (violates minimum notebooks) • Eliminate A
Test B) (15, 15): • n = 15 ≥ 10 ✓ • p = 15 < 20 ✗ (violates minimum pens) • Eliminate B
Test C) (20, 40): • n = 20 ≥ 10 ✓ • p = 40 ≥ 20 ✓ • 2(20) + 40 = 40 + 40 = 80 ≤ 100 ✓ • All constraints satisfied! ✓
Test D) (30, 50): • n = 30 ≥ 10 ✓ • p = 50 ≥ 20 ✓ • 2(30) + 50 = 60 + 50 = 110 ≤ 100 ✗ • Violates capacity constraint • Eliminate D
Answer: C) (20, 40)
SAT Tip: Check the "easier" constraints first (like minimum values) to quickly eliminate wrong answers.
4Problem 4medium
❓ Question:
A company produces notebooks and pens. Let n represent the number of notebooks and p represent the number of pens. The company must satisfy these constraints:
• Production capacity: 2n + p ≤ 100 • Minimum production: n ≥ 10 and p ≥ 20
Which point (n, p) represents a valid production plan?
A) (5, 30) B) (15, 15) C) (20, 40) D) (30, 50)
💡 Show Solution
We have three constraints to check:
- 2n + p ≤ 100 (capacity)
- n ≥ 10 (minimum notebooks)
- p ≥ 20 (minimum pens)
Test A) (5, 30): • n = 5 < 10 ✗ (violates minimum notebooks) • Eliminate A
Test B) (15, 15): • n = 15 ≥ 10 ✓ • p = 15 < 20 ✗ (violates minimum pens) • Eliminate B
Test C) (20, 40): • n = 20 ≥ 10 ✓ • p = 40 ≥ 20 ✓ • 2(20) + 40 = 40 + 40 = 80 ≤ 100 ✓ • All constraints satisfied! ✓
Test D) (30, 50): • n = 30 ≥ 10 ✓ • p = 50 ≥ 20 ✓ • 2(30) + 50 = 60 + 50 = 110 ≤ 100 ✗ • Violates capacity constraint • Eliminate D
Answer: C) (20, 40)
SAT Tip: Check the "easier" constraints first (like minimum values) to quickly eliminate wrong answers.
5Problem 5hard
❓ Question:
In the xy-plane, the solution set of the system of inequalities y > x² - 4 and y ≤ 6 - x is shown. A point with coordinates (a, b) lies in the solution set. Which of the following must be true?
A) b > a² - 4 only B) b ≤ 6 - a only C) b > a² - 4 AND b ≤ 6 - a D) b > a² - 4 OR b ≤ 6 - a
💡 Show Solution
If a point (a, b) is in the solution set of a SYSTEM of inequalities, it must satisfy ALL inequalities in the system.
The system is: • y > x² - 4 • y ≤ 6 - x
For point (a, b) to be in the solution set: • b > a² - 4 (first inequality with x = a, y = b) • AND b ≤ 6 - a (second inequality with x = a, y = b)
Both conditions must be true simultaneously.
Analyzing the options:
A) Only the first inequality - incomplete
B) Only the second inequality - incomplete
C) Both inequalities connected by AND - correct! ✓
D) Either inequality (OR) - this describes the UNION of two regions, not the intersection
Answer: C) b > a² - 4 AND b ≤ 6 - a
Key Concept: • Solution to a SYSTEM = intersection of all individual solution sets • Points must satisfy ALL inequalities (connected by AND) • This is different from "or" which would give the union
6Problem 6hard
❓ Question:
In the xy-plane, the solution set of the system of inequalities y > x² - 4 and y ≤ 6 - x is shown. A point with coordinates (a, b) lies in the solution set. Which of the following must be true?
A) b > a² - 4 only B) b ≤ 6 - a only C) b > a² - 4 AND b ≤ 6 - a D) b > a² - 4 OR b ≤ 6 - a
💡 Show Solution
If a point (a, b) is in the solution set of a SYSTEM of inequalities, it must satisfy ALL inequalities in the system.
The system is: • y > x² - 4 • y ≤ 6 - x
For point (a, b) to be in the solution set: • b > a² - 4 (first inequality with x = a, y = b) • AND b ≤ 6 - a (second inequality with x = a, y = b)
Both conditions must be true simultaneously.
Analyzing the options:
A) Only the first inequality - incomplete
B) Only the second inequality - incomplete
C) Both inequalities connected by AND - correct! ✓
D) Either inequality (OR) - this describes the UNION of two regions, not the intersection
Answer: C) b > a² - 4 AND b ≤ 6 - a
Key Concept: • Solution to a SYSTEM = intersection of all individual solution sets • Points must satisfy ALL inequalities (connected by AND) • This is different from "or" which would give the union
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