Solve Simple and Compound Linear Inequalities Practice

1. Solve for $x$.

   $12x-39\leq 9 \text{ AND }-4x+3$.
   

   Choose 1 answer:
   

   1. $x\geq4$
   2. $\dfrac{9}4$
   3. $x>-\dfrac{9}4$
   4. There are no solutions
      
   5. All values of $x$ are solutions

2. Solve for $x$.

   $12x+7< -11 \quad {\text{ OR }} \quad 5x-8>40$.
   

   Choose 1 answer:
   

   1. $x< -\dfrac{3}2$ or $x>\dfrac{48}{5}$
      
   2. $-\dfrac{3}2$
   3. $x>\dfrac{3}2$ or $x$
      
   4. There are no solutions
      
   5. All values of $x$ are solutions

3. Solve for $x$.

   $4x-39> -43 \quad {\text{ AND}} \quad 8x+31$.
   

   Choose 1 answer:
   

   1. $x$ or $x>-1$
      
   2. $x$
   3. $x>-1$
   4. There are no solutions
      
   5. All values of $x$ are solutions

4. Solve for $x$.

   $5x-4\geq 12 \quad {\text{ OR }} \quad 12x+5\leq-4$.
   

   Choose 1 answer:
   

   1. $x\geq\dfrac{16}{5}$ or $x\leq-\dfrac{3}{4}$
      
   2. $x\leq\dfrac{16}{5}$
   3. $x\geq-\dfrac{3}{4}$
      
   4. There are no solutions
      
   5. All values of $x$ are solutions

Source: Khan Academy, https://www.khanacademy.org/math/college-algebra/xa5dd2923c88e7aa8:linear-equations-and-inequalities/xa5dd2923c88e7aa8:compound-inequalities/e/compound_inequalities
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1. Understanding the type of statement

   We are given an $\text{AND}$ statement. A certain $x$‍-value is a solution of the statement if it satisfies both of the inequalities.
   

   Therefore, the solution of this statement is the intersection of the solutions of both inequalities. In other words, the overlap of the solutions of both inequalities is the solution of the statement.

   Finding the solutions to the two inequalities
   

   The solution of the first inequality, $12x-39\leq 9$, is ‍$x\leq 4$.

   The solution of the second inequality, $-4x+3 < -6$, is $x>\dfrac{9}4$.

   The inequalities ${x\leq4}$ and  ${x>\dfrac{9}4}$ are graphed below.

   
   

   The intersection of the solutions of both inequalities is $\dfrac{9}4 < x\leq4$.

   The answer

   The solution is $\dfrac{9}4 < x\leq4$.

   ---

2. Understanding the type of statement

   We are given an $\text{OR}$ statement. A certain $x$‍-value is a solution of the statement if it satisfies either of the inequalities.
   

   Therefore, the solution of this statement is the union of the solutions of both inequalities. In other words,the entire collection of the solutions of both inequalities forms the solution of this statement.

   Finding the solutions to the two inequalities
   

   The solution of the first inequality, $12x+7 < -11$, is ‍$x < -\dfrac{3}2$.

   The solution of the second inequality, $5x-8>40$, is $x>\dfrac{48}{5}$.

   The inequalities ${x < -\dfrac{3}2}$ and  ${x>\dfrac{48}{5}}$ are graphed below.
   

   
   

   
    
   
   

   The union of the solutions of both inequalities is all $x$-values that are less than $-\dfrac{3}2$ along with all ‍$x$-values that are greater than $\dfrac{48}5$.
   

   This can be represented mathematically as $x < -\dfrac{3}2$ OR $x>\dfrac{48}{5}$.

   The answer

   The solution is $x < -\dfrac{3}2$ OR $x>\dfrac{48}{5}$.

   ---

3. Understanding the type of statement

   We are given an $\text{AND}$ statement. A certain $x$‍-value is a solution of the statement if it satisfies both of the inequalities.
   

   Therefore, the solution of this statement is the intersection of the solutions of both inequalities. In other words, the overlap of the solutions of both inequalities is the solution of the statement.

   Finding the solutions to the two inequalities
   

   The solution of the first inequality, $4x-39 > -43$, is ‍$x\leq 4$.

   The solution of the second inequality, $8x+31 < 23$, is $x < -1$.

   The inequalities ${x > -1}$ and  ${x < -1}$ are graphed below.

   
   

   The intersection of the solutions of both inequalities is empty.

   The answer

   There are no solutions.

   ---

4. Understanding the type of statement

   We are given an $\text{OR}$ statement. A certain $x$‍-value is a solution of the statement if it satisfies either of the inequalities.
   

   Therefore, the solution of this statement is the union of the solutions of both inequalities. In other words,the entire collection of the solutions of both inequalities forms the solution of this statement.

   Finding the solutions to the two inequalities
   

   The solution of the first inequality, $5x-4\geq 12$, is ‍$x\geq \dfrac{16}{5}$.

   The solution of the second inequality, $12x+5\leq-4$, is $x\leq-\dfrac{3}{4}$.

   The inequalities ${x \geq \dfrac{16}{5}}$ and  ${x\leq-\dfrac{3}{4}}$ are graphed below.

   
   

   The union of the solutions of both inequalities is all $x$-values that are less than $\dfrac{16}5$ along with all ‍$x$-values that are greater than $-\dfrac{3}4$.
   

   This can be represented mathematically as $x\geq\dfrac{16}{5}$ OR $x\leq-\dfrac{3}{4}$.

   The answer

   The solution is $x\geq\dfrac{16}{5}$ OR $x\leq-\dfrac{3}{4}$.