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Quizzes > High School Quizzes > Electives

Practice Quiz: Linear Inequality Graphs

Practice solving graph-based inequality challenges with ease

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting a Compound Inequality Challenge quiz for high school algebra students.

Which of these compound inequalities best represents the statement 'x is greater than or equal to -2 and less than 4'?
x > -2 and x < 4
x ≥ -2 and x ≤ 4
x > -2 and x ≤ 4
x ≥ -2 and x < 4
The correct answer uses '≥' for -2 and '<' for 4, which exactly matches the description 'greater than or equal to -2 and less than 4'. The other options either reverse the inequality symbols or change the inclusion of endpoints.
What is the solution set for the compound inequality -1 < x ≤ 3?
[-1, 3]
[ -1, 3)
(-1, 3)
(-1, 3]
The inequality -1 < x ≤ 3 indicates that x is greater than -1 (not including -1) and less than or equal to 3 (including 3). The interval (-1, 3] is the correct representation of this situation.
When graphing a compound inequality on a number line, which symbol is used to indicate an endpoint that is not included?
Arrow
Open circle
Square bracket
Closed circle
An open circle is used to denote that an endpoint is not included in the solution set. A closed circle or square bracket would imply that the endpoint is part of the solution.
Which of the following represents the compound inequality 'x is less than 7 or x is greater than 10'?
x ≤ 7 or x ≥ 10
x ≤ 7 and x ≥ 10
x < 7 or x > 10
x < 7 and x > 10
The word 'or' indicates that x can satisfy either one of the conditions: x must be either less than 7 or greater than 10. Using 'and' would mistakenly require x to meet both conditions at the same time.
Solve the compound inequality: 2 ≤ x + 3 ≤ 8.
-1 ≤ x ≤ 5
-1 < x < 5
-5 ≤ x ≤ 1
-3 ≤ x ≤ 2
Subtracting 3 from all parts of the inequality gives -1 ≤ x ≤ 5, which is the correct solution set. The inclusion of -1 and 5 is essential based on the original inequality.
Determine the solution set of the compound inequality: 3x - 4 > 2 and 3x - 4 ≤ 8.
(2, 4)
(2, 4]
[2, 4]
[2, 4)
Solving the first inequality 3x - 4 > 2 gives x > 2, and the second inequality 3x - 4 ≤ 8 gives x ≤ 4. The intersection of these solutions is (2, 4] where 2 is not included and 4 is included.
Which property applies when multiplying or dividing both sides of an inequality by a negative number?
The inequality sign reverses
The inequality sign remains the same
Both sides become negative
The inequality is multiplied by -1
Multiplying or dividing an inequality by a negative number necessitates reversing the inequality sign to maintain a true statement. This is a fundamental property in solving inequalities.
For the inequality -2 ≤ 5 - x < 7, what is the solution for x?
(-2, 7]
[-2, 7]
[ -2, 7)
(-2, 7)
Starting with -2 ≤ 5 - x, subtracting 5 yields -7 ≤ -x, which flips to x ≤ 7. The inequality 5 - x < 7 gives x > -2. Together, these combine to form the interval (-2, 7].
Which graph best represents the solution set for the inequality x < -1 or x ≥ 3 on a number line?
Open circles at both -1 and 3 with shading in between
Closed circles at both -1 and 3 with shading in between
Closed circle at -1 left-shaded and open circle at 3 right-shaded
Open circle at -1 left-shaded and closed circle at 3 right-shaded
For x < -1, the graph shows an open circle at -1 with shading to the left. For x ≥ 3, a closed circle at 3 with shading to the right is used. Option A correctly describes this combined representation.
Solve and graph the inequality: 4 - 2x ≥ 0.
x > 2
x ≥ 2
x ≤ 2
x < 2
Rearranging the inequality 4 - 2x ≥ 0 leads to -2x ≥ -4; dividing by -2 (and flipping the inequality) gives x ≤ 2. This result is accurately reflected in Option A.
How is the phrase 'x is either less than -3 or greater than 5' expressed as a compound inequality?
x < -3 or x > 5
x ≤ -3 and x ≥ 5
x < -3 and x > 5
x ≤ -3 or x ≥ 5
The use of 'or' indicates that the solution set includes numbers that are either less than -3 or greater than 5, matching Option A correctly. Using 'and' would incorrectly imply that x satisfies both conditions simultaneously.
Which of the following inequalities is equivalent to 2 < 3 - x?
x < 1
x ≤ 1
x ≥ 1
x > 1
Subtracting 3 from both sides gives -1 < -x and multiplying both sides by -1 (and reversing the inequality) results in x < 1. Therefore, Option A is the correct equivalent form.
Select the correct description of the inequality |x - 2| < 3.
x is outside the interval [-1, 5]
x is between -1 and 5
x is greater than 3
x is less than 3
The absolute value inequality |x - 2| < 3 can be rewritten as -3 < x - 2 < 3, which after adding 2 to all parts becomes -1 < x < 5. This means x lies between -1 and 5.
What are the critical points of the compound inequality 0 ≤ 2x + 1 ≤ 7?
x = 0 and x = 7
x = -1 and x = 3.5
x = -0.5 and x = 3
x = -0.5 and x = 7
Setting 2x + 1 equal to the endpoints 0 and 7 gives x = -0.5 and x = 3 respectively. These numbers serve as the critical or boundary points for the inequality.
Find the solution to the inequality: 5 - 3x > 2x + 1.
x ≤ 0.8
x > 0.8
x ≥ 0.8
x < 0.8
Rearranging 5 - 3x > 2x + 1 leads to -5x > -4. Dividing by -5, and remembering to flip the inequality sign, yields x < 0.8. This is the correct solution for the inequality.
Solve the compound inequality: -2 < 3 - 2x ≤ 4.
x ≥ -0.5 and x < 2.5
x ≥ -0.5 and x ≤ 2.5
x > -0.5 and x < 2.5
x > -0.5 and x ≤ 2.5
Breaking the inequality into two parts: -2 < 3 - 2x leads to x < 2.5 and 3 - 2x ≤ 4 leads to x ≥ -0.5. Combining these gives the solution set x ≥ -0.5 and x < 2.5.
Which compound inequality represents the statement: 'x is not between 1 and 6 inclusive'?
x < 1 and x > 6
x > 1 and x < 6
x ≤ 1 or x ≥ 6
x < 1 or x > 6
The phrase 'not between 1 and 6 inclusive' means x lies outside the closed interval [1,6]. This is properly expressed as x < 1 or x > 6, where the 'or' signifies that satisfying either condition meets the requirement.
Given a number line with a closed circle at 2 and an open circle at 5, with shading in between, which compound inequality does it represent?
2 < x < 5
2 ≤ x < 5
2 ≤ x ≤ 5
2 < x ≤ 5
A closed circle at 2 indicates that 2 is included (x ≥ 2) while an open circle at 5 shows that 5 is not included (x < 5). Hence, the correct inequality is 2 ≤ x < 5.
Solve the inequality |2x - 3| ≥ 5.
x > -1 and x < 4
x ≥ -1 or x ≤ 4
x > -1 and x > 4
x ≤ -1 or x ≥ 4
An absolute value inequality of the form |A| ≥ k splits into two cases: A ≥ k or A ≤ -k. For |2x - 3| ≥ 5, solving gives 2x - 3 ≥ 5 (x ≥ 4) or 2x - 3 ≤ -5 (x ≤ -1).
Find the solution set for the inequality: (x + 2)/(x - 1) > 1.
x ≥ 1
x > 1
x < 1
x ≤ 1
Subtracting 1 from both sides gives (x + 2)/(x - 1) - 1 = 3/(x - 1) > 0. Since 3 is positive, the inequality holds when x - 1 > 0, resulting in x > 1. Note that x ≠1 due to the original denominator.
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Study Outcomes

  1. Identify compound inequalities represented by graphs.
  2. Analyze linear inequalities and their corresponding solution sets.
  3. Apply methods to solve compound inequalities accurately.
  4. Evaluate the consistency between graphical representations and algebraic solutions.
  5. Interpret results to confirm understanding of compound inequality concepts.

Compound & Linear Inequality Graph Cheat Sheet

  1. Compound inequalities overview - Compound inequalities combine two or more simple inequalities using "AND" or "OR" to define a specific range or union of solutions. They're a powerful way to express conditions like "between these values" or "either this range or that range." GeeksforGeeks
  2. Understanding "AND" - When you use "AND," both inequality conditions must be true at the same time, which means you're looking for the intersection of two solution sets. Picture overlapping regions on a number line - only the overlap counts! MathBitsNotebook
  3. Understanding "OR" - With "OR," at least one of the conditions needs to be true, giving you the union of solution sets. This is handy when you want to allow values in one range, another range, or both. MathBitsNotebook
  4. Graphing on a number line - Plot compound inequalities using open circles for strict inequalities (< or >) and closed circles for inclusive ones (≤ or ≥). Then shade the regions that satisfy your "AND" or "OR" statement to visualize the solution. OnlineMathLearning
  5. Step-by-step solving - Tackle each inequality separately by isolating the variable, then combine your results per the "AND" or "OR" rule. This methodical approach always leads you to the correct range or union. Symbolab
  6. Sign reversals - Never forget: multiplying or dividing both sides by a negative number flips the inequality sign! It's a small step, but missing it can completely change your solution set. MathBitsNotebook
  7. Interval notation - Express your final solutions concisely with interval notation, using brackets [ ] for inclusive ends and parentheses ( ) for exclusive ends. This shorthand is perfect for writing answers on tests or in reports. Symbolab
  8. Real‑world applications - Apply compound inequalities to real problems, like determining acceptable temperature ranges, dosage levels, or financial thresholds. Seeing them in action makes the concept stick! GeeksforGeeks
  9. Solution checking - Always plug a few test values back into your original inequalities to confirm they satisfy all conditions. This quick check helps you catch sign errors or mis-shaded regions. Symbolab
  10. Diverse practice - The more examples you tackle - both "AND" and "OR" types - the more intuitive solving and graphing compound inequalities become. Mix in word problems to challenge yourself and build confidence! OnlineMathLearning
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