Solve the inequality. Graph the solution set and write it in interval notation.
1/4 (x + 4) < 1/5 (3x + 2)
To solve the inequality, we will first simplify both sides:
1/4 (x + 4) < 1/5 (3x + 2)
Distribute the 1/4 and 1/5:
(x + 4)/4 < (3x + 2)/5
Multiply both sides by 4:
x + 4 < 4/5 (3x + 2)
Multiply both sides by 5:
5(x + 4) < 4 (3x + 2)
Distribute the 5 and 4:
5x + 20 < 12x + 8
Subtract 12x from both sides:
-7x + 20 < 8
Subtract 20 from both sides:
-7x < -12
Divide both sides by -7 (reversing the inequality):
x > 12/7
The solution set is all real numbers greater than 12/7.
Graphing this on a number line, we put an open circle at 12/7 and shade to the right to represent all the values greater than 12/7:
<=======o=========
12/7
To solve the inequality 1/4 (x + 4) < 1/5 (3x + 2), we need to simplify both sides and isolate the variable.
Let's start by simplifying the inequality:
1/4 (x + 4) < 1/5 (3x + 2)
Multiply both sides of the inequality by 20 (the least common multiple of 4 and 5) to eliminate the fractions:
20 * 1/4 (x + 4) < 20 * 1/5 (3x + 2)
5(x + 4) < 4(3x + 2)
5x + 20 < 12x + 8
Now, let's isolate the variable:
5x - 12x < 8 - 20
-7x < -12
Remember to divide by -7, but since we are dividing by a negative number, the inequality flips:
x > -12 / -7
x > 12/7
To graph the solution set, we represent the numbers on a number line. Since x is greater than 12/7, we indicate that the solution lies to the right of 12/7 on the number line.
Using interval notation, we can express the solution set as (12/7, +∞). This means that x is greater than 12/7 but not equal to it, going towards positive infinity.
Therefore, the solution set graphed on a number line and written in interval notation is (12/7, +∞).
To solve the inequality, we need to isolate the variable x.
Given: 1/4(x + 4) < 1/5(3x + 2)
First, we'll distribute the fractions on both sides of the inequality:
1/4 * x + 1/4 * 4 < 1/5 * 3x + 1/5 * 2
Simplifying further, we get:
1/4 * x + 1 < 3/5 * x + 2/5
Now, let's gather the x terms on one side of the inequality and the constant terms on the other side:
1/4 * x - 3/5 * x < 2/5 - 1
Combining the x terms and constant terms, we get:
(1/4 - 3/5) * x < 2/5 - 1
We can simplify further:
(-7/20) * x < -3/5
Now, we divide both sides by the coefficient of x, which is -7/20. Remember that when dividing by a negative number, the inequality sign flips:
x > (-3/5) / (-7/20)
Simplifying the right side:
x > (60/100) / (35/100)
x > 60/35
Simplifying the right side further:
x > 12/7
Now, we have the solution, x > 12/7. Let's represent this solution on a number line graph:
------o---------o------
12/7
The open circle indicates that 12/7 is not part of the solution since the inequality is strict (1/4(x + 4) is strictly less than 1/5(3x + 2)). The arrow to the right of the number line indicates values greater than 12/7.
In interval notation, we can write the solution as x ∈ (12/7, ∞). The interval (12/7, ∞) indicates all values of x greater than 12/7.
Write the answer in interval notation.
The answer in interval notation is (12/7, ∞).
Solve the inequality. Graph the solution set and write it in interval notation.
-6 < 2 (x - 3) <= 11
To solve the inequality, we will first simplify both sides:
-6 < 2(x - 3) <= 11
Distribute the 2:
-6 < 2x - 6 <= 11
Add 6 to all parts of the inequality:
0 < 2x <= 17
Divide all parts by 2 (reversing the inequality for negative numbers):
0/2 < 2x/2 <= 17/2
Simplify:
0 < x <= 17/2
The solution set is all real numbers greater than 0 and less than or equal to 17/2.
Graphing this on a number line, we put an open circle at 0 and a closed circle at 17/2, and shade between them to represent all the values greater than 0 and less than or equal to 17/2:
-------o=========o-----
0 17/2
In interval notation, we can express the solution set as (0, 17/2]. This means that x is greater than 0 and less than or equal to 17/2.
that was the incorrect answer
I apologize for the mistake. Let's redo the solution.
To solve the inequality, we will first simplify both sides:
-6 < 2(x - 3) ≤ 11
Distribute the 2:
-6 < 2x - 6 ≤ 11
Add 6 to all parts of the inequality:
0 < 2x ≤ 17
Divide all parts by 2:
0/2 < 2x/2 ≤ 17/2
Simplify:
0 < x ≤ 17/2
The solution set is all real numbers greater than 0 and less than or equal to 17/2.
Graphing this on a number line, we put an open circle at 0 and a closed circle at 17/2, and shade between them to represent all the values greater than 0 and less than or equal to 17/2:
-------o=========]
0 17/2
In interval notation, we can express the solution set as (0, 17/2]. This means that x is greater than 0 and less than or equal to 17/2.