Question

solve the inequalities.

1. ( 4t + 6 leq 10 )

( t leq 4 )

1. ( - 3n + 2 geq - 4n - 3 )
2. ( - 15 < 2x - 3 leq 13 )
3. solve ( |x| + 2 leq 6 ) and graph the solution.

Explanation:

Step1: Solve \(4t + 6\leq10\)

Subtract 6 from both sides:
\(4t+6 - 6\leq10 - 6\)
\(4t\leq4\)
Divide both sides by 4:
\(t\leq1\)

Step2: Solve \(-3n + 2\geq - 4n-3\)

Add \(4n\) to both sides:
\(-3n + 4n+2\geq-4n + 4n-3\)
\(n + 2\geq-3\)
Subtract 2 from both sides:
\(n+2 - 2\geq-3 - 2\)
\(n\geq-5\)

Step3: Solve \(-15\lt2x - 3\leq13\)

Add 3 to all parts:
\(-15 + 3\lt2x-3 + 3\leq13 + 3\)
\(-12\lt2x\leq16\)
Divide all parts by 2:
\(-6\lt x\leq8\)

Step4: Solve \(|x|+2\leq6\)

Subtract 2 from both sides:
\(|x|+2 - 2\leq6 - 2\)
\(|x|\leq4\)
This implies \(-4\leq x\leq4\). To graph, draw a number - line, put closed circles at \(x=-4\) and \(x = 4\), and shade the region between them.

Answer:

1. \(t\leq1\)
2. \(n\geq - 5\)
3. \(-6\lt x\leq8\)
4. \(-4\leq x\leq4\) (graph: closed circles at - 4 and 4, shaded between them)