Question

solve for ( x ).
( -15x + 4 leq 109 ) or ( -6x + 70 > -2 )
choose 1 answer:
a ( x geq -7 )
b ( -7 leq x < 12 )
c ( x < 12 )
d there are no solutions
e all values of ( x ) are solutions

Explanation:

Step1: Solve the first inequality \(-15x + 4 \leq 109\)

Subtract 4 from both sides: \(-15x \leq 109 - 4\)
Simplify: \(-15x \leq 105\)
Divide both sides by -15 (remember to reverse the inequality sign): \(x \geq \frac{105}{-15}\)
Simplify: \(x \geq -7\)

Step2: Solve the second inequality \(-6x + 70 > -2\)

Subtract 70 from both sides: \(-6x > -2 - 70\)
Simplify: \(-6x > -72\)
Divide both sides by -6 (reverse the inequality sign): \(x < \frac{-72}{-6}\)
Simplify: \(x < 12\)

Step3: Analyze the "OR" condition

The solution to \(x \geq -7\) OR \(x < 12\) includes all real numbers because every real number is either greater than or equal to -7, less than 12, or both. For example, a number like -8 is less than 12, a number like 13 is greater than or equal to -7, and a number like 5 is both. So all values of \(x\) satisfy at least one of the inequalities.

Answer:

E. All values of \(x\) are solutions