Answer :
Sure! Let's solve the inequality step by step and then graph the solution set on the number line.
Given the inequality:
[tex]\[ -4x + 3 \leq 6x - 57 \][/tex]
Step 1: Isolate the variable term on one side of the inequality
- First, we can add [tex]\(4x\)[/tex] to both sides to eliminate the [tex]\(-4x\)[/tex] term from the left side:
[tex]\[ 3 \leq 6x - 57 + 4x \][/tex]
[tex]\[ 3 \leq 10x - 57 \][/tex]
Step 2: Combine like terms
- Now, we combine the [tex]\(x\)[/tex] terms on the right side:
[tex]\[ 3 \leq 10x - 57 \][/tex]
Step 3: Isolate the variable term completely
- Next, add 57 to both sides to move the constant term to the left side:
[tex]\[ 3 + 57 \leq 10x \][/tex]
[tex]\[ 60 \leq 10x \][/tex]
Step 4: Solve for the variable
- Finally, divide both sides by 10 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{60}{10} \leq \frac{10x}{10} \][/tex]
[tex]\[ 6 \leq x \][/tex]
- This can also be written as:
[tex]\[ x \geq 6 \][/tex]
Interpretation:
- The solution to this inequality is [tex]\( x \geq 6 \)[/tex].
- On the number line, this is represented by a closed (filled-in) circle at [tex]\( x = 6 \)[/tex] and a line extending to the right, toward positive infinity.
So, the correct graph of the solution set shows a filled-in circle at [tex]\( 6 \)[/tex] and an arrow going to the right from [tex]\( 6 \)[/tex].
If we look at the provided choices, the correct answer will be the one that fits this description.
Given the inequality:
[tex]\[ -4x + 3 \leq 6x - 57 \][/tex]
Step 1: Isolate the variable term on one side of the inequality
- First, we can add [tex]\(4x\)[/tex] to both sides to eliminate the [tex]\(-4x\)[/tex] term from the left side:
[tex]\[ 3 \leq 6x - 57 + 4x \][/tex]
[tex]\[ 3 \leq 10x - 57 \][/tex]
Step 2: Combine like terms
- Now, we combine the [tex]\(x\)[/tex] terms on the right side:
[tex]\[ 3 \leq 10x - 57 \][/tex]
Step 3: Isolate the variable term completely
- Next, add 57 to both sides to move the constant term to the left side:
[tex]\[ 3 + 57 \leq 10x \][/tex]
[tex]\[ 60 \leq 10x \][/tex]
Step 4: Solve for the variable
- Finally, divide both sides by 10 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{60}{10} \leq \frac{10x}{10} \][/tex]
[tex]\[ 6 \leq x \][/tex]
- This can also be written as:
[tex]\[ x \geq 6 \][/tex]
Interpretation:
- The solution to this inequality is [tex]\( x \geq 6 \)[/tex].
- On the number line, this is represented by a closed (filled-in) circle at [tex]\( x = 6 \)[/tex] and a line extending to the right, toward positive infinity.
So, the correct graph of the solution set shows a filled-in circle at [tex]\( 6 \)[/tex] and an arrow going to the right from [tex]\( 6 \)[/tex].
If we look at the provided choices, the correct answer will be the one that fits this description.
