Question

solve the linear inequality for a. graph your answer on the number line. then write the solution in inequality notation.
-5a + 46a + 46 < 5(10a + 11)
show your work here
hint: to add inequalities (<, >, ≤, ≥), type \less\ or \greater\

Explanation:

Step1: Simplify both sides of the inequality

Combine like - terms on the left - hand side: \(-5a + 46a+46=( - 5 + 46)a+46 = 41a+46\).
Expand the right - hand side using the distributive property \(a(b + c)=ab+ac\): \(5(10a + 11)=5\times10a+5\times11 = 50a+55\).
The inequality becomes \(41a + 46<50a+55\).

Step2: Move the terms with \(a\) to one side

Subtract \(41a\) from both sides: \(41a+46-41a<50a + 55-41a\).
Simplify to get \(46<9a+55\).

Step3: Isolate the term with \(a\)

Subtract \(55\) from both sides: \(46-55<9a+55 - 55\).
\(-9<9a\).

Step4: Solve for \(a\)

Divide both sides by \(9\): \(\frac{-9}{9}<\frac{9a}{9}\).
Simplify to \(a>- 1\).

Answer:

The solution of the inequality \(-5a + 46a+46<5(10a + 11)\) is \(a>-1\). In inequality notation, it is \((-1,\infty)\). On the number line, we have an open circle at \(-1\) (because the inequality is strict, \(a
eq - 1\)) and the line extends to the right towards positive infinity.