Question

Solve the following inequalities and plot your answer on the number line:

a.) |7x-5| < 11
b.) |9-7x| <= 7

Answer 1

a.)
To solve |7x-5| < 11, first split the inequality into two cases based on the possible signs of 7x-5:
1) 7x-5 < 11
2) 7x-5 > -11

1) 7x-5 < 11
Add 5 to both sides:
7x < 16
Divide by 7:
x < 16/7

2) 7x-5 > -11
Add 5 to both sides:
7x > -6
Divide by 7:
x > -6/7

So, -6/7 < x < 16/7

b.)
To solve |9-7x| ≤ 7, first split the inequality into two cases based on the possible signs of 9-7x:
1) 9-7x ≤ 7
2) -(9-7x) ≤ 7

1) 9-7x ≤ 7
Subtract 9 from both sides:
-7x ≤ -2
Divide by -7 (switching signs):
x ≥ 2/7

2) -(9-7x) ≤ 7
Distribute the negative sign:
-9 + 7x ≤ 7
Add 9 to both sides:
7x ≤ 16
Divide by 7:
x ≤ 16/7

So, 2/7 ≤ x ≤ 16/7

Plotting these on the number line, we have:

<-------------------------------------o---|---o------------------------------------>
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12

The interval for the first inequality is (-6/7, 16/7) and the interval for the second inequality is [2/7, 16/7].