Question

1. solve for ( x ) and graph the solution on the number line below. if possible, resolve your answer to a single inequality. in case of no solution (( varnothing )), leave the number line blank.

( 5x + 10 > 30 ) or ( 5x + 10 > 70 )

Explanation:

Step1: Simplify first inequality

Subtract 10 from both sides:
$5x + 10 - 10 > 30 - 10$
$5x > 20$
Divide by 5:
$\frac{5x}{5} > \frac{20}{5}$
$x > 4$

Step2: Simplify second inequality

Subtract 10 from both sides:
$5x + 10 - 10 > 70 - 10$
$5x > 60$
Divide by 5:
$\frac{5x}{5} > \frac{60}{5}$
$x > 12$

Step3: Combine "or" inequalities

For an "or" compound inequality, the solution includes all values that satisfy either inequality. Since all values greater than 12 are already included in values greater than 4, the combined solution is the broader set.

Step4: Graph the solution

On the number line, draw an open circle at 4, then shade all values to the right of 4.

<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|--->
   -12  -10   -8   -6   -4   -2    0    2    4    6    8   10   12
                          (===============================>

(Open circle at 4, arrow pointing right)

Answer:

The simplified single inequality is $x > 4$. The number line has an open circle at 4 and is shaded to the right of 4.