How do you solve the inequality: #5x + 10 >=10# and #7x - 7 <=14#?

1 Answer
Apr 15, 2018

Answer:

#" "#
Required Solution: #color(blue)(0<=x<=3#

Interval Notation: #color(blue)([ 0,3 ]#

Explanation:

#" "#
The Compound Inequality Expression given:

#color(red)(5x+10>=10 and 7x-7<=14#

#color(green)("Step 1"#

Consider #5x+10>=10# first and simplify.

Subtract #color(red)(10# to both sides of the inequality to obtain

#5x+10- color(red)(10) >=10 - color(red)(10)#

#5x+cancel 10- color(red)(cancel 10) >=cancel 10 - color(red)(cancel 10)#

#5x>=0#

Divide both sides of the inequality by #color(red)(5)#

#(5x)/color(red)(5)>=0/color(red)(5)#

#(cancel 5x)/color(red)(cancel 5)>=0/color(red)(5)#

#color(blue)(x>=0# Intermediate Solution 1

#color(green)("Step 2"#

Consider #7x-7<=14# next.

Add #color(red)(7)# to both sides of the inequality to get

#7x-7+color(red)(7)<=14+color(red)(7)#

#7x-cancel 7+color(red)(cancel 7)<=14+color(red)(7)#

#7x<=21#

Divide both sides of the inequality by #color(red)(7#

#(7x)/color(red)(7)<=21/color(red)(7#

#(cancel 7x)/color(red)(cancel 7)<=cancel 21^color(red)3/color(red)(cancel 7#

#color(blue)(x<=3# Intermediate Solution 2

#color(green)("Step 3"#

Combine both the Intermediate Solution to obtain:

#color(blue)(x>=0 and color(blue)(x<=3#

#color(red)(0<=x<=3# The required solution

Using the interval notation: #color(red)([0,3]#

#color(green)("Step 4"#

You can verify the results using a graph

The first graph below is created using the two inequality expressions given:

#color(red)(5x+10>=10 and 7x-7<=14#

The solution is the shaded common region

enter image source here

You can also graph just the solution to obtain a solution graph:

#color(red)(0<=x<=3#

enter image source here

Hope you find this solution useful.