Start by decomposing the compound inequalities and simplifying all individual inequalities. Notice that some signs are strict so interpret them carefully.
a. C b. A c. D d. B
Practice makes perfect
We need to match the compound inequalities with the corresponding graphs for their solutions. We will work each case individually.
Inequality A
The first given inequality is -26 < 6x - 8 < 16, which is a compound inequality. Thus, it can be decomposed into two individual inequalities.
-26 < 6x - 8 and 6x - 8 < 16
We can simplify each individual inequality by isolating x. We can achieve this by working with it as if it were an equation. We will use inverse operations. Let's first simplify -26 < 6x - 8.
Therefore, the solution set for Inequality A contains all numbers greater than -3 ANDless than 4, not including 4 and -3 themselves.
x> -3 AND x < 4
The graph showing this solution set is Graph b.
Inequality B
We now have the inequality - 13 ≤ x + 23 ≤ 2. Again, we are working with a compound inequality. Let's then decompose it and simplify each individual inequality. We can write it as an AND inequality.
\begin{gathered}
\text{-} \dfrac{1}{3} \leq \dfrac {x + 2}{3} \quad \textbf{AND} \quad \dfrac {x + 2}{3} \leq 2
\end{gathered}
We will now simplify - 13 ≤ x + 23.
Therefore, the solution set of Inequality B contains all numbers greater than or equal to -3, andless than or equal to 4.
x ≥ - 3 AND x ≤ 4
The graph showing this solution set is Graph d.
Inequality C
We have the inequality 4x+1<-11 OR x2 -5 > - 3. Let's simplify each individual inequality. We will start with 4x + 1 < - 11.
Therefore, the solution set for Inequality C contains all numbers which are greater than 4 orless than -3. Note that 4 and -3 are not included in the solution set.
x < - 3 OR x > 4
The graph showing this solution set is Graph a.
Inequality D
Finally we have x -63 ≤ -3 OR 2x + 8≥ 16. Let's simplify each individual inequality. We will start with x -63 ≤ -3.
The solution set for Inequality D contains all numbers which are less than or equal to -3 ORgreater than or equal to 4.
x ≤ - 3 OR x ≥ 4
The graph showing this solution set is Graph c.
