Question Video: Solving Compound Inequalities Mathematics

Find all the values of 𝑥 that satisfy 5𝑥 < 15 or 7𝑥 + 3 ≥ 52.

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Video Transcript

Find all the values of 𝑥 that satisfy five 𝑥 is less than 15 or seven 𝑥 plus three is greater than or equal to 52.

In order to answer this question, we will begin by solving each of the inequalities individually. Firstly, we have five 𝑥 is less than 15. If we divide both sides of this inequality by five, we have 𝑥 is less than three. The second inequality states that seven 𝑥 plus three is greater than or equal to 52. We can subtract three from both sides, giving us seven 𝑥 is greater than or equal to 49. Dividing by seven, we have 𝑥 is greater than or equal to seven.

We now have two solutions, and we want either one of these to be correct. This can be demonstrated on a number line. For the solution 𝑥 is less than three, we have an open circle at three and 𝑥 can take any value to the left of this. As 𝑥 is greater than or equal to seven, we have a solid circle at seven and 𝑥 can take this value or any value to the right of this.

We can therefore conclude that the values of 𝑥 that satisfy five 𝑥 is less than 15 or seven 𝑥 plus three is greater than or equal to 52 are 𝑥 is less than three or 𝑥 is greater than or equal to seven. Another way of describing this would be that 𝑥 can take any real value apart from those that are greater than or equal to three and less than seven.

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