Graph the inequality on the numberline and then write it in interval notation.
6x+5(x+5)≥7x+45
and
12x+4(x-3)<3x+53
Please help me with these equation and how to write the solution on a numberline. Thanks.
6x+5(x+5)≥7x+45
11 x + 25 ≥ 7 x + 45
4 x ≥ 20
x ≥ 5
black dot at x = 5, arrow to right
12 x + 4 x -12 < 3 x + 53
13 x < 65
x < 65/3
open circle at x = 65/3 and arrow to left
What would the interval notation be for
6x+5(x+5)≥7x+45
x ≥ 5
Thank You Damon for the assistance with interval notation and graphing. I am horrible at this section.
To graph the inequality on a number line and write it in interval notation, we need to solve the inequality for x. Let's solve each equation step by step:
1) 6x + 5(x + 5) ≥ 7x + 45
First, distribute the 5 through the parenthesis:
6x + 5x + 25 ≥ 7x + 45
Combine like terms on both sides of the inequality:
11x + 25 ≥ 7x + 45
Next, subtract 7x from both sides:
4x + 25 ≥ 45
Then, subtract 25 from both sides:
4x ≥ 20
Finally, divide both sides by 4:
x ≥ 5
Now let's graph the inequality on a number line:
Draw a horizontal number line and label it with integers. Mark the point x = 5 with a closed or filled-in circle (since it includes the value of 5). Then, shade to the right of the circle to represent all values greater than or equal to 5. This signifies the solutions to the inequality.
To write the solution in interval notation, we use square brackets [ ] for closed endpoints and parentheses ( ) for open endpoints. The inequality x ≥ 5 includes the value 5, so we use a closed bracket on 5. As the inequality continues indefinitely to the right, we use the open interval (5, ∞) to represent all values greater than 5.
Thus, the solution is written in interval notation as (5, ∞).
2) 12x + 4(x - 3) < 3x + 53
First, distribute the 4 through the parenthesis:
12x + 4x - 12 < 3x + 53
Combine like terms on both sides of the inequality:
16x - 12 < 3x + 53
Next, subtract 3x from both sides:
13x - 12 < 53
Then, add 12 to both sides:
13x < 65
Finally, divide both sides by 13:
x < 5
Now let's graph the inequality on a number line:
Draw a horizontal number line and label it with integers. Mark the point x = 5 with an open circle (since it does not include the value of 5). Then, shade to the left of the circle to represent all values less than 5. This signifies the solutions to the inequality.
To write the solution in interval notation, we use square brackets [ ] for closed endpoints and parentheses ( ) for open endpoints. The inequality x < 5 does not include the value 5, so we use an open parenthesis on 5. As the inequality continues indefinitely to the left, we use the open interval (-∞, 5) to represent all values less than 5.
Thus, the solution is written in interval notation as (-∞, 5).