Linear Inequalities
If \( x+y = 7\) and \( 4 \leq x \leq 19,\) what is the range of \(y?\)
If the inequality \(a^2(x+1)>576 x+24a\) holds for all real numbers \(x,\) what is the constant \(a?\)
Let \(x>\frac{14}{13}\) be the solution to the inequality \(bx-(a+b)<0,\) where \(a\) and \(b\) are constants. Then what is the solution to the inequality \(ax+2a+b<0?\)
A factory produces two products \(A\) and \(B\) using raw materials \(p\) and \(q\). It uses \(3\) units of \(p\) and \(1\) unit of \(q\) to produce one unit of \(A\). It uses \(1\) unit of \(p\) and \(2\) units of \(q\) to produce one unit of \(B\). The maximum daily supplies are \(30\) units of \(p\) and \(20\) units of \(q.\) If the profit per unit of \(A\) is \(\$3\) and the profit per unit of \(B\) is \(\$2\), what is the maximum total daily profit (in dollars)?
Real numbers \(x\) and \(y\) satisfy \( -3 < x < 1 \) and \( -1 < y < 5\). What is the range of \( x - y ? \)