Question Video: Solving Compound Inequalities Mathematics

Video Transcript

Find all the values of 𝑥 that satisfy the compound inequality seven 𝑥 minus five is greater than negative 12 or six 𝑥 plus three is greater than or equal to 15.

We will begin by solving each of the inequalities in turn. Firstly, we have seven 𝑥 minus five is greater than negative 12. Adding five to both sides gives us seven 𝑥 is greater than negative seven as negative 12 plus five equals negative seven. We can then divide both sides of this inequality by seven such that 𝑥 is greater than negative one.

The second inequality states that six 𝑥 plus three is greater than or equal to 15. We begin by subtracting three from both sides so that six 𝑥 is greater than or equal to 12. We can then divide both sides of this inequality by six giving us 𝑥 is greater than or equal to two.

The keyword in the question here is “or.” We don’t need both of the inequalities to be true, just one of them. We can represent the first inequality on a number line as shown. We have an open circle at negative one, and 𝑥 can take any value greater than this. The second solution, 𝑥 is greater than or equal to two, can be represented by a solid circle at the value two. 𝑥 can take any value equal to or greater than this.

If 𝑥 is greater than or equal to two, it is also greater than negative one. This means that the values of 𝑥 that satisfy the compound inequality seven 𝑥 minus five is greater than negative 12 or six 𝑥 plus three is greater than or equal to 15 is 𝑥 is greater than negative one.