Find all the values of x that satisfy at least one of the...

Key Concepts

Logical Operators in Inequalities (Or)

The logical operator 'or' in inequalities indicates that the overall solution includes all values that meet at least one of the individual conditions. Understanding how this operator works is key, as it dictates whether to merge solution ranges or restrict them based on overlapping conditions.

Union of Solution Sets

The union of solution sets refers to combining the solutions of individual conditions such that any element belonging to at least one of the sets is included in the final solution. In the context of inequalities connected by 'or', this ensures that all possible satisfying values are considered.

Linear Inequalities

Linear inequalities involve expressions where variables are compared using inequality symbols such as >, <, ?, or ?. Solving these requires manipulation similar to equations, while carefully observing rules that govern inequalities, such as the reversal of the inequality sign when multiplying or dividing by a negative number.

Compound Inequalities

Compound inequalities combine two or more inequalities. When they are linked by 'or', the solution set is the union of the individual solutions; that is, it includes any values that satisfy at least one of the inequalities. This concept contrasts with 'and', where only values satisfying every inequality are accepted.

Transcript

00:01 To find the values of x that satisfy at least one of the two inequalities, we have a.

00:12 We have 2 to x minus 7 greater than 1.

00:23 Or, okay, 2.

00:29 So what we can do is let's add 7 to both sides plus 7.

00:36 So i have 2x minus 7 plus 7 greater than 1 plus 7.

00:45 And this will give me 2x greater than 8.

00:54 So from here we divide both sides by 2.

01:00 And i have x greater than 4.

01:08 Or the other parts we have 2x plus 1 greater than 3 so all we represent by this symbol so i add minus 1 to both sides so i have 2x plus 1 minus 1 less than 3 minus 1 and this one and this will give me 2 x less than two so you divide both sides by two and you have x to be less than one so this implies that's the interval of solution or the values of x you are looking at from here you have four greater than four so to infinity union so this is union.

02:22 This will be less than 1, so to negative infinity.

02:29 Then for b, b, we have 2x minus 7 to be less than or equal to 1.

02:49 So let's add plus 7 to both sides so that i have 2x minus 7 plus 7 plus 7.

02:58 7 less than or equal to 1 plus 7.

03:04 So this will give me 2x less than or equal to 8.

03:14 Then divide 2 by 2.

03:18 So i have x less than or equal to 4.

03:24 If you take the other parts, which is all, so all we represent by v, i have 2x plus 1 less than 3.

03:38 We add minus 1 to both sides.

03:43 So i have 2x plus 1 minus 1 less than 3 minus 1.

03:53 And this gives me 2x less than 2.

03:58 So let's divide both sides by 2...