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Number - Compound and double inequalities

Grade 10IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A compound inequality consists of two inequalities joined by the words 'and' or 'or'.

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A double inequality is a type of 'and' inequality written in the form a<x<ba < x < b. This means that xx is greater than aa AND xx is less than bb simultaneously.

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When solving a double inequality like a≤f(x)≤ba \leq f(x) \leq b, any operation performed to isolate the variable must be applied to all three parts (the left, the middle, and the right).

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Critical Rule: When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed (<< becomes >>, and ≤\leq becomes ≥\geq).

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On a number line, an open circle ∘\circ represents 'greater than' (>>) or 'less than' (<<), meaning the endpoint is excluded. A closed circle ∙\bullet represents 'greater than or equal to' (≥\geq) or 'less than or equal to' (≤\leq), meaning the endpoint is included.

📐Formulae

a<x<b  ⟺  x>a and x<ba < x < b \iff x > a \text{ and } x < b

If c<0 and a<b, then ac>bc\text{If } c < 0 \text{ and } a < b, \text{ then } ac > bc

If c>0 and a<b, then ac<bc\text{If } c > 0 \text{ and } a < b, \text{ then } ac < bc

x∈[a,b]  ⟺  a≤x≤bx \in [a, b] \iff a \leq x \leq b

💡Examples

Problem 1:

Solve the double inequality: −7≤2x−3<5-7 \leq 2x - 3 < 5.

Solution:

−7≤2x−3<5-7 \leq 2x - 3 < 5 Add 33 to all three parts: −7+3≤2x−3+3<5+3-7 + 3 \leq 2x - 3 + 3 < 5 + 3 −4≤2x<8-4 \leq 2x < 8 Divide all three parts by 22: −42≤2x2<82\frac{-4}{2} \leq \frac{2x}{2} < \frac{8}{2} −2≤x<4-2 \leq x < 4

Explanation:

To solve a double inequality, we isolate the variable xx in the middle by performing the same inverse operations on all sections of the inequality.

Problem 2:

Solve for xx: −12<−3(x+1)≤9-12 < -3(x + 1) \leq 9.

Solution:

First, divide all parts by −3-3. Since we are dividing by a negative number, we must flip the inequality signs: −12−3>−3(x+1)−3≥9−3\frac{-12}{-3} > \frac{-3(x + 1)}{-3} \geq \frac{9}{-3} 4>x+1≥−34 > x + 1 \geq -3 Subtract 11 from all parts: 4−1>x≥−3−14 - 1 > x \geq -3 - 1 3>x≥−43 > x \geq -4 Rearrange to standard form: −4≤x<3-4 \leq x < 3

Explanation:

When dividing by the negative coefficient −3-3, the signs reverse. The final step rearranges the inequality so the smaller number is on the left.

Problem 3:

Solve the compound inequality: 3x+1<−53x + 1 < -5 or 2x−4>62x - 4 > 6.

Solution:

Solve the first inequality: 3x+1<−53x + 1 < -5 3x<−63x < -6 x<−2x < -2 Solve the second inequality: 2x−4>62x - 4 > 6 2x>102x > 10 x>5x > 5 The combined solution is: x<−2 or x>5x < -2 \text{ or } x > 5

Explanation:

For an 'or' compound inequality, we solve each inequality separately. The solution set includes all values that satisfy either one of the conditions. This is represented on a number line as two separate rays pointing in opposite directions.