Question

use the number lines provided to graph the solution set for the inequalities. things to keep in mind: did you solve for the variable? should the sign flip or stay the same? will you use a circle or a dot on the value? did you check your number line by testing out a value? ( 3x geq x - 8 ) ( \frac { x + 5 } { 2 } < 4 ) ( 6 - 3x leq \frac { 6x + 42 } { 4 } )

Explanation:

Step1: Solve \(3x\geq x - 8\)

Subtract \(x\) from both sides: \(3x-x\geq x - 8-x\), which gives \(2x\geq - 8\).
Divide both sides by \(2\): \(x\geq - 4\). On the number - line, we use a closed dot at \(-4\) (since the inequality is \(\geq\)) and shade to the right.

Step2: Solve \(\frac{x + 5}{2}<4\)

Multiply both sides by \(2\): \(x + 5<8\).
Subtract \(5\) from both sides: \(x<3\). On the number - line, we use an open circle at \(3\) (since the inequality is \(<\)) and shade to the left.

Step3: Solve \(6-3x\leq\frac{6x + 42}{4}\)

Multiply both sides by \(4\) to get \(24-12x\leq6x + 42\).
Add \(12x\) to both sides: \(24\leq18x + 42\).
Subtract \(42\) from both sides: \(24-42\leq18x\), so \(-18\leq18x\).
Divide both sides by \(18\): \(x\geq - 1\). On the number - line, we use a closed dot at \(-1\) (since the inequality is \(\leq\)) and shade to the right.

Answer:

For \(3x\geq x - 8\), the solution is \(x\geq - 4\).
For \(\frac{x + 5}{2}<4\), the solution is \(x<3\).
For \(6-3x\leq\frac{6x + 42}{4}\), the solution is \(x\geq - 1\).