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) $x=8$ $$-1<\frac{3-x}{2} \leq 1$$

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) $x=8$
$$-1<\frac{3-x}{2} \leq 1$$

Key Concepts

Compound Inequalities

Compound inequalities involve two or more inequalities that a solution must satisfy at the same time. In the context of the given problem, the inequality -1 < (3 - x)/2 ? 1 sets two simultaneous conditions that must hold true for any valid solution. This requires checking that both parts of the inequality are satisfied for each candidate value.

Substitution Method

The substitution method is a technique used to determine whether a particular value satisfies an equation or inequality by replacing the variable with that value. For this problem, each candidate value of x is substituted into the inequality to verify if the resulting statement is true, thereby confirming or rejecting the value as a solution.

Inequality Manipulation

Inequality manipulation involves the process of performing algebraic operations on inequalities to simplify them, while taking care to reverse the inequality sign when multiplying or dividing by negative values. This is essential for solving linear inequalities and understanding the range of solutions, or for reformatting inequalities to a more testable form.

Linear Inequalities

Linear inequalities are expressions involving a linear function of a variable set in relation to another quantity using inequality symbols. They represent a range of possible solutions on the number line rather than a specific value. In this context, the inequality -1 < (3 - x)/2 ? 1 is a linear inequality that must be handled with standard techniques for both solving and testing candidate solutions.

Transcript

00:02 Given the inequality, negative 1 is less than 3 minus x divided by 2 is less than or equal to 1.

00:08 We're going to determine whether or not x equals negative 1, x equals square width of 3, x equals 1, and x equals 5 are solutions to this inequality.

00:18 We're going to do that through substitution and seeing if the statement we get is in fact true.

00:23 So for x equals negative 1, we get negative 1 is less than 3 minus negative 1 over 2 is less than or equal to 1.

00:32 1.

00:33 Negative 1 is less than 3 minus negative 1 is 3 plus 1.

00:37 So we get 4 over 2 is less than or equal to 1.

00:42 Negative 1 is less than 2 is less than or equal to 1.

00:46 Negative 1 is in fact less than 2.

00:48 However, it is not less than or equal to 1.

00:51 So negative 1 does not work as a solution to this inequality.

00:55 Our second one is x equals the square of 3.

00:58 Negative 1 is less than 3 minus the square to 3 over 2 is less than or equal to 1.

01:06 Negative 1 is less than 3 minus the square to 3 is roughly 0 .76 over 2 is less than or equal to 1.

01:17 Negative 1 is less than 0 .76 divided by 2 is 0 .38 is less than or equal to 1.

01:26 Negative 1 is in fact less than 0 .38 and 0 .38 is in fact less than or equal to 1...

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