Question

determine whether or not the following statements are true or false. explain your reasoning for each result and write a corrected true statement if it is false.
a) (x < 3) and (3 < x) mean the same thing since inequalities work the same as an equal sign.
b) (-4x>12) is the same as (x < - 3) since dividing both sides of the inequality requires a flip of the inequality symbol.
c) (-3(x + 2)leq5) is the same as (-3x-6geq5) since multiplying one side of the inequality by a negative requires a flip of the inequality symbol.
d) (3x + 11geq5) is the same as (3xleq - 6) since adding a negative number to both sides of an inequality requires a flip of the inequality symbol.

Explanation:

Step1: Analyze statement a

$x < 3$ means $x$ is less than 3, while $3 < x$ means $x$ is greater than 3. They are not the same. Inequalities are directional and not symmetric like the equal - sign. So it's false. The corrected statement is: $x < 3$ and $3 < x$ do not mean the same thing.

Step2: Analyze statement b

Starting with $-4x>12$, when we divide both sides by - 4, according to the rule of inequalities (dividing by a negative number flips the inequality sign), we get $x < - 3$. So this statement is true.

Step3: Analyze statement c

Expand $-3(x + 2)$ using the distributive property: $-3(x + 2)=-3x-6$. The original inequality is $-3(x + 2)\leq5$, which is $-3x - 6\leq5$, not $-3x - 6\geq5$. The sign - flip rule applies when multiplying or dividing the entire inequality by a negative number, not just when distributing. So it's false. The corrected statement is: $-3(x + 2)\leq5$ is the same as $-3x - 6\leq5$.

Step4: Analyze statement d

Starting with $3x + 11\geq5$, subtract 11 from both sides: $3x+11 - 11\geq5 - 11$, which gives $3x\geq - 6$. Adding or subtracting a number (positive or negative) to both sides of an inequality does not flip the inequality sign. Only multiplying or dividing by a negative number flips the sign. So it's false. The corrected statement is: $3x + 11\geq5$ is the same as $3x\geq - 6$.

Answer:

a) False. Corrected: $x < 3$ and $3 < x$ do not mean the same thing.
b) True.
c) False. Corrected: $-3(x + 2)\leq5$ is the same as $-3x - 6\leq5$.
d) False. Corrected: $3x + 11\geq5$ is the same as $3x\geq - 6$.