Determine whether each value of x is a solution of the...

Determine whether each value of $x$ is a solution of the inequality. (Values) (a) $x=3 \quad$ (b) $x=-3$ (c) $x=\frac{5}{2} \quad$ (d) $x=\frac{3}{2}$ (a) $x=-\frac{1}{2} \quad$ (b) $x=-\frac{5}{2}$ (c) $x=\frac{4}{3} \quad$ (d) $x=0$ (a) $x=-1 \quad$ (b) $x=\sqrt{5}$ (c) $x=1 \quad$ (d) $x=5$ (a) $x=13 \quad$ (b) $x=-1$ (c) $x=14 \quad$ (d) $x=8$ $$-5<2 x-1 \leq 1$$

Determine whether each value of $x$ is a solution of the inequality.
(Values)
(a) $x=3 \quad$ (b) $x=-3$
(c) $x=\frac{5}{2} \quad$ (d) $x=\frac{3}{2}$
(a) $x=-\frac{1}{2} \quad$ (b) $x=-\frac{5}{2}$
(c) $x=\frac{4}{3} \quad$ (d) $x=0$
(a) $x=-1 \quad$ (b) $x=\sqrt{5}$
(c) $x=1 \quad$ (d) $x=5$
(a) $x=13 \quad$ (b) $x=-1$
(c) $x=14 \quad$ (d) $x=8$
$$-5<2 x-1 \leq 1$$

Key Concepts

Testing Candidate Solutions

Testing candidate solutions is the method of substituting specific values into an inequality to check whether they satisfy the condition. This is useful for verifying potential solutions, especially when dealing with compound inequalities where each candidate must meet multiple criteria.

Understanding Boundary Values

Boundary values represent the endpoints in the range of values that satisfy an inequality. It is important to determine whether these endpoint values are included in the solution set, which depends on whether the inequality is strict (using '<' or '>') or inclusive (using '?' or '?').

Algebraic Manipulation of Inequalities

The process of solving inequalities involves performing algebraic operations—such as addition, subtraction, multiplication, and division—on both sides of the inequality. A crucial aspect of this concept is that the direction of the inequality must be reversed when multiplying or dividing both sides by a negative number.

Linear Inequalities

Linear inequalities involve expressions with variables raised only to the first power and can be manipulated using similar rules as linear equations, with additional care given to the direction of the inequality sign. This is fundamental for determining the range of values that satisfy the given condition.

Compound Inequalities

Compound inequalities combine two inequality statements and require both conditions to be satisfied at the same time. This concept involves understanding the intersection of two solution sets, where each part of the inequality must hold true for any value of the variable to be considered a solution.

Transcript

00:02 Given the inequality, negative 5 is less than 2x minus 1 is less than or equal to 1.

00:08 We're going to determine whether the solutions negative 1⁄2, negative 5 -half, 4 thirds, and 0 are solutions to this inequality.

00:17 So in our first one here, we have negative 5, it's less than 2 times, and i'm going to substitute in that negative 1⁄2.

00:24 It's greater than or equal to 1.

00:26 Be it negative 5 is less than 2 times negative 1 half is negative 1 minus 1 is greater than or equal to 1.

00:34 Negative 5 is less than negative 2, less than or equal to 1.

00:40 So what we're looking at here is negative 5 and 1 are my endpoints.

00:45 It needs to be between there.

00:46 It can't equal negative 5, but it can equal negative 1.

00:49 And anything between those two endpoints is a solution.

00:53 Negative 2 is in fact between those two points.

00:58 Negative 2 or negative 5 is less than negative 2 and negative 2 is less than or equal to 1, which makes this a true statement and negative 1 half is a solution to this inequality.

01:11 Our next one we get negative 5 is less than 2 times negative 5 halves minus 1.

01:18 It's greater, less than or equal to 1.

01:20 Negative 5 less than.

01:23 We've got two times negative five halves, so my twos are going to cancel.

01:27 And we get negative 5 minus 1 is less than or equal to 1.

01:31 We get negative 5 is less than negative 6 is less than or equal to 1.

01:37 Negative 6 is not less than negative 6...

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