Write an inequality and solve.
Two times the sum of a number and four is no more than three times the sum of the number and seven decreased by four.
2(x+4) <= 3(x+7)-4
...
The sum of a number times and is at most .
Let's start by representing the description of the problem with variables.
Let's assume the number is represented by the variable 'x'.
According to the problem, the inequality can be written as:
2(x + 4) ≤ 3(x + 7) - 4
Now let's solve the inequality step-by-step:
2(x + 4) ≤ 3(x + 7) - 4
2x + 8 ≤ 3x + 21 - 4
2x + 8 ≤ 3x + 17
2x - 3x + 8 ≤ 3x - 3x + 17
-x + 8 ≤ 17
-x ≤ 17 - 8
-x ≤ 9
Now, let's multiply both sides of the inequality by -1 to isolate 'x' and to change the direction of the inequality:
-x(-1) ≥ 9(-1)
x ≥ -9
Therefore, the solution to the inequality is x ≥ -9.
Let's break down the problem step by step:
Step 1: Define the unknown variable:
Let's say the unknown number is "x".
Step 2: Translate the given information into an inequality:
"Two times the sum of a number and four is no more than three times the sum of the number and seven decreased by four."
Translated inequality: 2(x + 4) ≤ 3(x + 7) - 4
Step 3: Simplify the inequality:
2x + 8 ≤ 3x + 21 - 4
Step 4: Combine like terms:
2x + 8 ≤ 3x + 17
Step 5: Move all terms containing "x" to one side and the constant terms to the other side of the inequality:
2x - 3x ≤ 17 - 8
-x ≤ 9
Step 6: Divide both sides of the inequality by -1 (note: since we are dividing by a negative number, the inequality sign flips):
x ≥ -9
Step 7: Interpret the solution:
The solution to the inequality is x ≥ -9, which means the unknown number has to be greater than or equal to -9.