Question Video: Solving Absolute Value Inequalities Algebraically | Nagwa Question Video: Solving Absolute Value Inequalities Algebraically | Nagwa

Question Video: Solving Absolute Value Inequalities Algebraically Mathematics • Second Year of Secondary School

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Find the solution set of the inequality |𝑥 − 3| ≤ 7.

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Video Transcript

Find the solution set of the inequality the absolute value of 𝑥 minus three is less than or equal to seven.

We will solve this inequality both algebraically and graphically. If the absolute value of an expression is less than or equal to a positive number, then either the expression is less than or equal to this number, or the expression is greater than or equal to the negative of this number. In this question, we need to solve the two inequalities 𝑥 minus three is less than or equal to seven and 𝑥 minus three is greater than or equal to negative seven.

Adding three to both sides of our first inequality gives us 𝑥 is less than or equal to 10. Repeating this for the second inequality gives us 𝑥 is greater than or equal to negative four. This can be demonstrated on a number line. 𝑥 is less than or equal to 10 and greater than or equal to negative four. Writing this as a solution set, we have the closed interval from negative four to 10.

As previously mentioned, we can also demonstrate this graphically by firstly drawing the coordinate 𝑥- and 𝑦-axes. The straight line 𝑦 equals 𝑥 minus three has 𝑦-intercept at negative three and slope or gradient equal to one. We need to draw the absolute value of this which will be V shaped. The portion of the graph below the 𝑥-axis will be reflected in this line such that the 𝑦-intercept is now equal to three.

Drawing the horizontal line 𝑦 is equal to seven intersects the graph at two points, this gives us our two solutions of negative four and 10. The portion of the graph where the absolute value of 𝑥 minus three is less than or equal to seven lies between negative four and 10 inclusive. This confirms that the solution set negative four to 10 is correct.

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