solve the compound inequalities as demostrated being careful of how a negative x- term is handled in the solving process show all math work arriving at the solutions display a simple line qraph for each solution set as demostrated. -1<3+2x<11 (Your and compound inequality) 1-x>-3 or 5x-1>19 (your or Copmound inequality)
-1<3+2x<11
subtract 3
-4 < 2x < 8
divide by 2
-2 < x < 4
1-x ≥ -3 OR 5x-1 > 19
-x ≥ -4 OR 5x > 20
x ≤ 4 OR x > 4
which obviously includes every value of x
To solve the compound inequality -1 < 3 + 2x < 11, we first need to solve the inequality separately for the upper and lower bounds. Let's start with the lower bound:
-1 < 3 + 2x
To isolate the variable x, we will subtract 3 from all parts of the inequality:
-1 - 3 < 3 + 2x - 3
Simplifying further:
-4 < 2x
Next, we divide both sides of the inequality by 2 (the coefficient of x), remembering to reverse the inequality since we are dividing by a negative number:
-4/2 > 2x/2
-2 > x
Now, let's move on to the upper bound:
3 + 2x < 11
By subtracting 3 from all parts of the inequality, we get:
3 + 2x - 3 < 11 - 3
Simplifying further:
2x < 8
Dividing both sides of the inequality by 2 gives:
2x/2 < 8/2
x < 4
So, we have found the lower bound x > -2 and the upper bound x < 4. Combining these inequalities, the solution set is -2 < x < 4.
To display this solution set on a number line graph, you would draw a line with an open circle at -2 and another open circle at 4. Then, shade everything between the two open circles to represent the values within the solution set.
Now, let's move on to the compound inequality 1 - x > -3 or 5x - 1 > 19.
Let's solve the first inequality 1 - x > -3:
To isolate the variable x, we subtract 1 from all parts of the inequality:
1 - x - 1 > -3 - 1
Simplifying further:
-x > -4
Multiplying both sides of the inequality by -1 (remembering to reverse the inequality) gives:
x < 4
Now, let's solve the second inequality 5x - 1 > 19:
To isolate the variable x, we add 1 to all parts of the inequality:
5x - 1 + 1 > 19 + 1
Simplifying further:
5x > 20
Dividing both sides of the inequality by 5 gives:
5x/5 > 20/5
x > 4
So, we have found x < 4 from the first inequality and x > 4 from the second inequality. Combining these inequalities, the solution set is x < 4 or x > 4.
To display this solution set on a number line graph, you would draw an open circle at 4 and shade everything to the left and right of the open circle to represent the values within the solution set.