Question

multi-step inequalities
solve each inequality and graph its solution.

1. ( 3 < - 5 n + 2 n )
2. ( - p - 4 p > - 10 )
3. ( 9 geq - 2 m + 2 - 3 )
4. ( 6 - 4 ( 6 n + 7 ) geq 122 )
5. ( 167 < 6 + 7 ( 2 - 7 r ) )
6. ( - 8 x + 2 x - 16 < - 5 x + 7 x )

Explanation:

Step1: Combine like terms

For \(3 < -5n + 2n\), combine \(-5n\) and \(2n\) to get \(3<-3n\).
For \(-p - 4p>-10\), combine \(-p\) and \(-4p\) to get \(-5p > - 10\).
For \(9\geq-2m + 2-3\), simplify \(2 - 3=-1\), so \(9\geq-2m-1\).
For \(6-4(6n + 7)\geq122\), use the distributive property \(a(b + c)=ab+ac\): \(6-(24n+28)\geq122\), then \(6 - 24n-28\geq122\), and combine \(6-28=-22\), so \(-24n-22\geq122\).
For \(167 < 6+7(2 - 7r)\), use the distributive property: \(167<6+(14 - 49r)\), then \(167<6 + 14-49r\), and combine \(6 + 14 = 20\), so \(167<20-49r\).
For \(-8x + 2x-16<-5x + 7x\), combine \(-8x+2x=-6x\) and \(-5x + 7x = 2x\), so \(-6x-16<2x\).

Step2: Isolate the variable term

For \(3<-3n\), divide both sides by \(- 3\). Remember that when dividing by a negative number, the inequality sign flips. So \(n<-1\).
For \(-5p > - 10\), divide both sides by \(-5\) (flip the inequality sign), we get \(p < 2\).
For \(9\geq-2m-1\), add \(1\) to both sides: \(9 + 1\geq-2m\), so \(10\geq-2m\).
For \(-24n-22\geq122\), add \(22\) to both sides: \(-24n\geq122 + 22\), so \(-24n\geq144\).
For \(167<20-49r\), subtract \(20\) from both sides: \(167-20<-49r\), so \(147<-49r\).
For \(-6x-16<2x\), add \(6x\) to both sides: \(-16<2x + 6x\), so \(-16<8x\).

Step3: Solve for the variable

For \(10\geq-2m\), divide both sides by \(-2\) (flip the inequality sign), \(m\geq - 5\).
For \(-24n\geq144\), divide both sides by \(-24\) (flip the inequality sign), \(n\leq - 6\).
For \(147<-49r\), divide both sides by \(-49\) (flip the inequality sign), \(r<-3\).
For \(-16<8x\), divide both sides by \(8\), \(x>-2\).

Answer:

1. \(n < - 1\)
2. \(p < 2\)
3. \(m\geq - 5\)
4. \(n\leq - 6\)
5. \(r < - 3\)
6. \(x>-2\)

To graph:

• For \(n < - 1\): Draw an open circle at \(-1\) and shade to the left.
• For \(p < 2\): Draw an open circle at \(2\) and shade to the left.
• For \(m\geq - 5\): Draw a closed circle at \(-5\) and shade to the right.
• For \(n\leq - 6\): Draw a closed circle at \(-6\) and shade to the left.
• For \(r < - 3\): Draw an open circle at \(-3\) and shade to the left.
• For \(x > - 2\): Draw an open circle at \(-2\) and shade to the right.