Question
Solve and graph the solution set. \[ 6 x+6+x<5+3 x-3 \]
Ask by Cole Owen. in the United States
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( x < -1 \). On the number line, this is represented by shading all numbers to the left of -1, with an open circle at -1.
Solution
Step 1. Combine like terms.
On the left side, combine 6x and x to get 7x.
The inequality becomes:
7x + 6 < 5 + 3x - 3.
Step 2. Simplify the right side.
Subtract 3 from 5:
5 - 3 = 2,
so the inequality is:
7x + 6 < 3x + 2.
Step 3. Isolate the variable.
Subtract 3x from both sides:
7x - 3x + 6 < 2,
4x + 6 < 2.
Subtract 6 from both sides:
4x < 2 - 6,
4x < -4.
Divide both sides by 4:
x < -1.
Solution Set: x < -1.
Graphing the solution:
Draw a number line and mark the point -1 with an open circle (indicating -1 is not included). Shade the line to the left of -1 to show that all numbers less than -1 are part of the solution.
Graph on the Number Line:
←===========( -1 )––––––––––––→
This represents all x such that x < -1.
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Simplify this solution The Deep Dive
To solve the inequality \( 6x + 6 + x < 5 + 3x - 3 \), first, combine like terms: \( 7x + 6 < 3x + 2 \). Next, subtract \( 3x \) from both sides: \( 4x + 6 < 2 \). Now, subtract 6 from both sides: \( 4x < -4 \). Finally, divide by 4: \( x < -1 \). To graph the solution set, draw a number line, and place an open circle at -1 to indicate that -1 is not included in the solution. Shade the area to the left of -1, representing all numbers less than -1. Now we have both the solution and the graphical representation!
