GCE Mathematics Questions Solved Inequalities.

Welcome to today’s tutorial!
In this lesson, we’ll solve some of the most frequently asked questions in GCE Mathematics, including inequalities and completing the square. These topics are common in exam papers and essential for building a strong foundation in algebra.

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You can watch the full class in the video below:

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Question 1: Solving Inequality Equations

Problem:
Solve the inequality: 3x+7≥±23x + 7 \geq \pm 23x+7≥±2

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Step-by-Step Solution:

We are asked to solve: 3x+7≥2or3x+7≥−23x + 7 \geq 2 \quad \text{or} \quad 3x + 7 \geq -23x+7≥2or3x+7≥−2

Let’s solve each part:

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First part: 3x+7≥2⇒3x≥2−7⇒3x≥−5⇒x≥−533x + 7 \geq 2 \Rightarrow 3x \geq 2 – 7 \Rightarrow 3x \geq -5 \Rightarrow x \geq \frac{-5}{3}3x+7≥2⇒3x≥2−7⇒3x≥−5⇒x≥3−5​

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Second part: 3x+7≥−2⇒3x≥−2−7⇒3x≥−9⇒x≥−31=−33x + 7 \geq -2 \Rightarrow 3x \geq -2 – 7 \Rightarrow 3x \geq -9 \Rightarrow x \geq \frac{-3}{1} = -33x+7≥−2⇒3x≥−2−7⇒3x≥−9⇒x≥1−3​=−3

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Final Solution:

x≥−53orx≥−3x \geq \frac{-5}{3} \quad \text{or} \quad x \geq -3x≥3−5​orx≥−3

The set of values satisfying the inequality includes all values greater than or equal to −53\frac{-5}{3}3−5​, since that is more restrictive than −3-3−3.

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Question 2: Completing the Square

Problem:
Write the quadratic expression x2−4x+12x^2 – 4x + 12×2−4x+12

in the form: (x+a)2+b(x + a)^2 + b(x+a)2+b

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Step-by-Step Using the Completing the Square Method:

1. Start with the expression:

x2−4x+12x^2 – 4x + 12×2−4x+12

2. Move the constant term to the right side:

x2−4x=−12x^2 – 4x = -12×2−4x=−12

3. Take half of the coefficient of xxx, which is −4-4−4, divide by 2 to get −2-2−2, and square it:

(−2)2=4(-2)^2 = 4(−2)2=4

4. Add this square (4) to both sides:

x2−4x+4=−12+4⇒(x−2)2=−8x^2 – 4x + 4 = -12 + 4 \Rightarrow (x – 2)^2 = -8×2−4x+4=−12+4⇒(x−2)2=−8

5. Bring the constant back to the expression form:

(x−2)2+8(x – 2)^2 + 8(x−2)2+8

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Final Answer:

x2−4x+12=(x−2)2+8x^2 – 4x + 12 = (x – 2)^2 + 8×2−4x+12=(x−2)2+8

This is now in the required completed square form.

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Watch more educational videos and past questions: https://youtube.com/@allcbts

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Why These Topics Are Important

These types of questions — solving inequalities and completing the square — are crucial in:

• Solving quadratic equations
• Graphing parabolas
• Optimization problems
• JAMB, GCE, and WAEC exams

Mastering them boosts your confidence and improves your speed during exams.

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Learn how to solve top GCE Mathematics questions step-by-step. Covers inequality equations and completing the square using exam-standard methods.

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