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2 <p>Last updated on<strong>October 28, 2025</strong></p>

3 <p>A quadratic inequality compares a quadratic expression ax² + bx + c (where a ≠ 0) to zero using >, <, ≥, ≤ signs. They allow us to analyze intervals between roots and help determine if a given condition holds. Students should rewrite the inequality in standard form for solving, find the roots of ax² + bx + c = 0, and test sign intervals.</p>

4 <h2>What are Quadratic Inequalities?</h2>

5 <p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>

6 <p>▶</p>

7 <ul><li>\(x^2 + x - 1 > 0 \)</li>

8 </ul><ul><li>\(2x^2 - 5x - 2 \)</li>

9 </ul><ul><li>\(x^2 + 2x - 1 < 0 \)</li>

10 </ul><h2>What are the Types of Quadratic Inequalities?</h2>

11 <p>Quadratic inequalities compare a<a>quadratic</a><a>expression</a>to zero, using inequality<a>symbols</a>:<a>greater than</a>,<a>less than</a>, greater than or equal to, and less than or equal to.</p>

12 <p>They are solved by finding roots and testing the signs of the expression over different intervals.</p>

13 <strong>Type</strong><strong>General Form</strong><strong>Name</strong>Standard \(ax^2 + bx + c ≷ 0\) Positive/Negative/0m/0 No real roots \(<0\) Always positive or negative Repeated roots \(=0 \) Touches the x-axis once Factored (simple/repeated) \((x - a)(x - b) . . .≷0\) Linear and repeated linear<a>factors</a>Rational \(\frac{P(x)}{Q(x)} \gtrless 0 \) Rational quadratic inequalities Compound quad ≷ m quad Two-sided/bounded inequality<h2>How To Solve Quadratic Inequalities</h2>

14 <p><strong>Step 1 - Rewrite the equation to express the inequality:</strong></p>

15 <p>Example: \(x^2 - 5x + 6 > 0\)</p>

16 <p><strong>Step 2 - Find roots by factoring or<a>formula</a>:</strong></p>

17 <p>\((x-2)(x-3)>0\) → roots: 2, 3.</p>

18 <p><strong>Step 3 - Create intervals:</strong></p>

19 <p>\((-∞,2), (2,3), (3,∞)\)</p>

20 <p><strong>Step 4 - Test each interval by plugging in a sample value:</strong></p>

21 <ul><li>\(x = 0\): positive → true </li>

22 <li>\(x = 2.5 \): negative → false </li>

23 <li>\(x = 4 \): positive → true</li>

24 </ul><p><strong>Step 5. Write a solution:</strong></p>

25 <p>All \(x∈ (-∞,2) (3,∞) \)</p>

26 <h3>Explore Our Programs</h3>

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28 <h2>Notations Used In Quadratic Inequalities.</h2>

27 <h2>Notations Used In Quadratic Inequalities.</h2>

29 <p>The symbols > and < (greater than, less than) replace “=” in a quadratic equation to form a quadratic inequality. The format is \(ax² + bx + c > 0 \) or \(ax² + bx + c < 0\).</p>

28 <p>The symbols > and < (greater than, less than) replace “=” in a quadratic equation to form a quadratic inequality. The format is \(ax² + bx + c > 0 \) or \(ax² + bx + c < 0\).</p>

30 <ul><li><strong>( ) →</strong>Open brackets</li>

29 <ul><li><strong>( ) →</strong>Open brackets</li>

31 </ul><ul><li><strong>[ ] →</strong>Closed brackets</li>

30 </ul><ul><li><strong>[ ] →</strong>Closed brackets</li>

32 </ul><ul><li><strong>o →</strong>Open value (x cannot take this value)</li>

31 </ul><ul><li><strong>o →</strong>Open value (x cannot take this value)</li>

33 </ul><ul><li><strong>• →</strong>Closed value (x can take this value)</li>

32 </ul><ul><li><strong>• →</strong>Closed value (x can take this value)</li>

34 </ul><p>Examples:</p>

33 </ul><p>Examples:</p>

35 <ul><li><strong>(-1, 1) →</strong>x cannot take -1 and 1</li>

34 <ul><li><strong>(-1, 1) →</strong>x cannot take -1 and 1</li>

36 </ul><ul><li><strong>[-1, 1) →</strong>x can take -1 but not 1</li>

35 </ul><ul><li><strong>[-1, 1) →</strong>x can take -1 but not 1</li>

37 </ul><ul><li><strong>(-1, 1] →</strong>x cannot take -1 but can take 1</li>

36 </ul><ul><li><strong>(-1, 1] →</strong>x cannot take -1 but can take 1</li>

38 </ul><ul><li><strong>[-1, 1] →</strong>x can take both -1 and 1</li>

37 </ul><ul><li><strong>[-1, 1] →</strong>x can take both -1 and 1</li>

39 </ul><h2>What are the methods to solve Quadratic Inequalities?</h2>

38 </ul><h2>What are the methods to solve Quadratic Inequalities?</h2>

40 <p>Different methods used to solve quadratic inequalities are listed below:</p>

39 <p>Different methods used to solve quadratic inequalities are listed below:</p>

41 <h3>Factoring + Test Intervals</h3>

40 <h3>Factoring + Test Intervals</h3>

42 <ol><li>Rewrite as \(ax² + bx + c ≷ 0\). </li>

41 <ol><li>Rewrite as \(ax² + bx + c ≷ 0\). </li>

43 <li>Factor & find roots. </li>

42 <li>Factor & find roots. </li>

44 <li>Divide the<a>number line</a>into intervals. </li>

43 <li>Divide the<a>number line</a>into intervals. </li>

45 <li>Test each interval.</li>

44 <li>Test each interval.</li>

46 </ol><p><strong>Example:</strong></p>

45 </ol><p><strong>Example:</strong></p>

47 <p>\(x^2 - 5x + 6 > 0 ⟹ (x - 2)(x - 3) > 0\)</p>

46 <p>\(x^2 - 5x + 6 > 0 ⟹ (x - 2)(x - 3) > 0\)</p>

48 <p>\((-∞,2), (2,3), (3,∞) \)</p>

47 <p>\((-∞,2), (2,3), (3,∞) \)</p>

49 <p>Testing the values in the expression, we get, \(x = 0 \): true;\( x = 2.5 \): false; \(x = 4 \): true.</p>

48 <p>Testing the values in the expression, we get, \(x = 0 \): true;\( x = 2.5 \): false; \(x = 4 \): true.</p>

50 <p><strong>Solution:</strong>\(x∈(-∞,2)∪(3,∞) \)</p>

49 <p><strong>Solution:</strong>\(x∈(-∞,2)∪(3,∞) \)</p>

51 <h3>Sign Chart Method</h3>

50 <h3>Sign Chart Method</h3>

52 <ol><li>Factor the expression. </li>

51 <ol><li>Factor the expression. </li>

53 <li>Mark the roots on a number line. </li>

52 <li>Mark the roots on a number line. </li>

54 <li>Determine the sign of each factor in each region (alternating sign from rightmost interval). </li>

53 <li>Determine the sign of each factor in each region (alternating sign from rightmost interval). </li>

55 <li>Combine to find where the<a>product</a>matches the inequality.</li>

54 <li>Combine to find where the<a>product</a>matches the inequality.</li>

56 </ol><p><strong>Example:</strong></p>

55 </ol><p><strong>Example:</strong></p>

57 <p>\(\frac{P(x)}{Q(x)} \gtrless 0 \)</p>

56 <p>\(\frac{P(x)}{Q(x)} \gtrless 0 \)</p>

58 <p>\(x - 3 = 0, x = 3 \)</p>

57 <p>\(x - 3 = 0, x = 3 \)</p>

59 <p>\(x + 2 = 0, x = -2 \)</p>

58 <p>\(x + 2 = 0, x = -2 \)</p>

60 <p>So, the expression changes sign between \(x = -2\) and \(x = 3\)</p>

59 <p>So, the expression changes sign between \(x = -2\) and \(x = 3\)</p>

61 <p>Now,<a>set</a>intervals using roots to divide the number line:</p>

60 <p>Now,<a>set</a>intervals using roots to divide the number line:</p>

62 <p>\((-∞,-2), (-2,3), (3,∞)\)</p>

61 <p>\((-∞,-2), (-2,3), (3,∞)\)</p>

63 <p>Test each interval</p>

62 <p>Test each interval</p>

64 <p>\(x = -3: (x - 3) (x + 2) = (+)\)</p>

63 <p>\(x = -3: (x - 3) (x + 2) = (+)\)</p>

65 <p>\(x = 0: (x - 3) (x + 2) = (-) \)</p>

64 <p>\(x = 0: (x - 3) (x + 2) = (-) \)</p>

66 <p>\(x = 4: (x - 3) (x + 2) = (+)\)</p>

65 <p>\(x = 4: (x - 3) (x + 2) = (+)\)</p>

67 <p>The only region where the expression is negative is \((-2, 3)\) So, \(x∈(-2,3)\)</p>

66 <p>The only region where the expression is negative is \((-2, 3)\) So, \(x∈(-2,3)\)</p>

68 <p><strong>Solution:</strong>\(x∈(-2,3)\)</p>

67 <p><strong>Solution:</strong>\(x∈(-2,3)\)</p>

69 <h3>Graphical Method</h3>

68 <h3>Graphical Method</h3>

70 <ol><li>Rewrite as \(y = ax2 + bx + c and y ≷ 0\) </li>

69 <ol><li>Rewrite as \(y = ax2 + bx + c and y ≷ 0\) </li>

71 <li>Sketch the parabola (upwards if \(a>0\), down if \(a<0\)) </li>

70 <li>Sketch the parabola (upwards if \(a>0\), down if \(a<0\)) </li>

72 <li>Read off where the parabola is above/below the x-axis.</li>

71 <li>Read off where the parabola is above/below the x-axis.</li>

73 </ol><p><strong>Example:</strong></p>

72 </ol><p><strong>Example:</strong></p>

74 <p>\(x^2 + 5x + 6 ≥ 0 \)</p>

73 <p>\(x^2 + 5x + 6 ≥ 0 \)</p>

75 <p>The graph crosses at \(x = -2\) and \(-3\) and opens upward since the linear<a>coefficient</a>is positive.</p>

74 <p>The graph crosses at \(x = -2\) and \(-3\) and opens upward since the linear<a>coefficient</a>is positive.</p>

76 <p>So, the expression is \(≥ 0\) outside \([-3,-2]\).</p>

75 <p>So, the expression is \(≥ 0\) outside \([-3,-2]\).</p>

77 <p><strong>Solution:</strong>\(x∈(-∞, -3]∪[-2,∞)\).</p>

76 <p><strong>Solution:</strong>\(x∈(-∞, -3]∪[-2,∞)\).</p>

78 <h2>Tips and Tricks to Master Quadratic Inequalities</h2>

77 <h2>Tips and Tricks to Master Quadratic Inequalities</h2>

79 <p>The given tips and tricks help students understand and work with quadratic inequalities efficiently. </p>

78 <p>The given tips and tricks help students understand and work with quadratic inequalities efficiently. </p>

80 <ul><li><p><strong>Factorize first</strong>- Convert the quadratic equation into its<a>factored form</a>to easily find the roots. </p>

79 <ul><li><p><strong>Factorize first</strong>- Convert the quadratic equation into its<a>factored form</a>to easily find the roots. </p>

81 </li>

80 </li>

82 <li><p><strong>Mark critical points</strong>- Plot the roots on a number line to divide intervals for testing signs. </p>

81 <li><p><strong>Mark critical points</strong>- Plot the roots on a number line to divide intervals for testing signs. </p>

83 </li>

82 </li>

84 <li><p><strong>Test intervals</strong>- Check the sign of each interval to determine where the inequality holds true. </p>

83 <li><p><strong>Test intervals</strong>- Check the sign of each interval to determine where the inequality holds true. </p>

85 </li>

84 </li>

86 <li><p><strong>Remember parabola direction</strong>- Use the coefficient of \(x^2\) to know if the parabola opens upward or downward. </p>

85 <li><p><strong>Remember parabola direction</strong>- Use the coefficient of \(x^2\) to know if the parabola opens upward or downward. </p>

87 </li>

86 </li>

88 <li><p><strong>Use clear notation</strong>- Write your final answer in interval form or inequality form for clarity.</p>

87 <li><p><strong>Use clear notation</strong>- Write your final answer in interval form or inequality form for clarity.</p>

89 </li>

88 </li>

90 </ul><h2>Real-Life Applications of the Quadratic Inequalities</h2>

89 </ul><h2>Real-Life Applications of the Quadratic Inequalities</h2>

91 <p>The quadratic inequalities model helps us understand and solve real-world problems in the fields of physics, engineering, economics, and biology. Some of its applications are listed below:</p>

90 <p>The quadratic inequalities model helps us understand and solve real-world problems in the fields of physics, engineering, economics, and biology. Some of its applications are listed below:</p>

92 <ul><li><strong>Bridge design</strong>: This is useful to ensure the structural integrity of bridges. For example, civil engineers need to examine the maximum load a bridge can bear to determine the range of weight distribution that is safe. This helps in preventing breakdown.</li>

91 <ul><li><strong>Bridge design</strong>: This is useful to ensure the structural integrity of bridges. For example, civil engineers need to examine the maximum load a bridge can bear to determine the range of weight distribution that is safe. This helps in preventing breakdown.</li>

93 </ul><ul><li><strong>Profit Margins in Business:</strong>Businesses use Quadratic Inequalities for modelling and analyzing the business<a>profit</a>margins. For example, a company uses it to decide the range of product prices that will create a desired profit level.</li>

92 </ul><ul><li><strong>Profit Margins in Business:</strong>Businesses use Quadratic Inequalities for modelling and analyzing the business<a>profit</a>margins. For example, a company uses it to decide the range of product prices that will create a desired profit level.</li>

94 </ul><ul><li><strong>Radio Telescope Design:</strong>Radio telescopes use parabolic mirrors to focus incoming radio waves. This helps design the shape and dimensions of these mirrors to ensure optimal signal reception and focusing. </li>

93 </ul><ul><li><strong>Radio Telescope Design:</strong>Radio telescopes use parabolic mirrors to focus incoming radio waves. This helps design the shape and dimensions of these mirrors to ensure optimal signal reception and focusing. </li>

95 </ul><ul><li><strong>Determining Safe Speed:</strong>This is useful in transportation, as it helps to find the Maximum safe speed on a curved and dangerous road map. Making sure that the vehicle can navigate the curve without skidding or losing control.</li>

94 </ul><ul><li><strong>Determining Safe Speed:</strong>This is useful in transportation, as it helps to find the Maximum safe speed on a curved and dangerous road map. Making sure that the vehicle can navigate the curve without skidding or losing control.</li>

96 </ul><ul><li><strong>Area Calculation:</strong>This can be used to find the dimensions of a shape, like a rectangle, that meets a specific area requirement. For example, to find the dimensions of a garden or a plot that has the minimum required area.</li>

95 </ul><ul><li><strong>Area Calculation:</strong>This can be used to find the dimensions of a shape, like a rectangle, that meets a specific area requirement. For example, to find the dimensions of a garden or a plot that has the minimum required area.</li>

97 </ul><h2>Common Mistakes of the Quadratic Inequalities and How to Avoid Them</h2>

96 </ul><h2>Common Mistakes of the Quadratic Inequalities and How to Avoid Them</h2>

98 <p>Quadratic inequalities are challenging for some students, and common mistakes can create incorrect answers. Recognizing and avoiding these errors is important for perfect problem-solving.</p>

97 <p>Quadratic inequalities are challenging for some students, and common mistakes can create incorrect answers. Recognizing and avoiding these errors is important for perfect problem-solving.</p>

98 + <h2>Download Worksheets</h2>

99 <h3>Problem 1</h3>

100 <p>For which values of x is x² - 4x + 3 > 0?</p>

101 <p>Okay, lets begin</p>

102 <p>x∈ (-∞,1) ∪ (3,∞).</p>

103 <h3>Explanation</h3>

104 <p>(x - 1)(x - 3) 3 > 0. Roots: x = 1, 3.</p>

105 <p>(-∞,1), (1,3), (3,∞).</p>

106 <p>x = 0: (0 - 1)(0 - 3) = (-1)(-3) 3 > 0, true</p>

107 <p>x = 2: (2 - 1)(2 - 3) = 1 · (-1) = -1 > 0, false</p>

108 <p>X = 4: (4 - 1)(4 - 3) = 3·1 = 3 > 0, true</p>

109 <p>Well explained 👍</p>

110 <h3>Problem 2</h3>

111 <p>Solve x² + 2x - 15 ≤ 0.</p>

112 <p>Okay, lets begin</p>

113 <p>-5 ≤ x ≤ 3.</p>

114 <h3>Explanation</h3>

115 <p>Factor to (x + 5)(x - 3).</p>

116 <p>The quadratic is non-positive between the roots.</p>

117 <p>Well explained 👍</p>

118 <h3>Problem 3</h3>

119 <p>Determine x such that 3x² - 12 ≤ 0.</p>

120 <p>Okay, lets begin</p>

121 <p>-2 ≤ x ≤ 2</p>

122 <h3>Explanation</h3>

123 <p>Rewrite as 3(x2 - 4) ≤ 0.</p>

124 <p>Hence, -2 ≤ x ≤ 2.</p>

125 <p>Well explained 👍</p>

126 <h3>Problem 4</h3>

127 <p>Find all x satisfying 2x²+ 7x + 6 < 0</p>

128 <p>Okay, lets begin</p>

129 <p>-2 < x < -3/2</p>

130 <h3>Explanation</h3>

131 <p>Factor (2x + 3)(x + 2) < 0.</p>

132 <p>The product is negative between the roots: -2 < x < -1.5.</p>

133 <p>Well explained 👍</p>

134 <h3>Problem 5</h3>

135 <p>Solve 4x² - 4x+ 1 ≥ 0.</p>

136 <p>Okay, lets begin</p>

137 <p>All real x.</p>

138 <h3>Explanation</h3>

139 <p>This is (2x - 1)2, which is always ≥ 0 for every x.</p>

140 <p>Well explained 👍</p>

141 <h2>FAQs of the Quadratic Inequalities</h2>

142 <h3>1.What is a quadratic inequality?</h3>

143 <p>A quadratic inequality is an inequality involving a quadratic expression, such as ax2 + bx + c, where the inequality symbol is <, ≤, >, or ≥.</p>

144 <h3>2.How do I solve a quadratic inequality?</h3>

145 <p>To solve, factor the quadratic expression, find the roots, determine intervals, test points within those intervals, and identify where the inequality holds.</p>

146 <h3>3.What is the significance of the sign of a in ax² + bx + c?</h3>

147 <p>The sign of a determines the direction of the parabola: if a > 0, it opens upwards; if a < 0, it opens downwards.</p>

148 <h3>4.How do I handle strict inequalities < or >?</h3>

149 <p>For strict inequalities, the solution does not include the boundary points; use open intervals to represent the solution set.</p>

150 <h3>5.How do I express the solution set?</h3>

151 <p>Solution sets can be expressed in<a>interval notation</a>, such as (-∞,-3)∪(2,∞), or in set-builder notation.</p>

152 <h3>6.How can I help my child understand quadratic inequalities at home?</h3>

153 <p>Use graphs to show how a parabola lies above or below the x-axis for different values of 𝑥 x. Visualizing helps your child see which regions satisfy the inequality.</p>

154 <h3>7.What are some real-life examples of quadratic inequalities I can use to explain the concept?</h3>

155 <p>Examples include finding safe speed limits for turns, determining profit ranges in business, or calculating projectile heights that stay below a certain level.</p>

156