We have now looked at solving inequalities that involve one step or two steps to solve. We're now going to take a look at how we can use inequalities to solve problems given a written description.
Much as with solving equations from worded problems, there are certain keywords or phrases to look out for. When it comes to inequalities, we now have a few extra keywords and phrases to represent the different inequality symbols.
Construct and solve an inequality for the following situation:
"The sum of $2$2 groups of $x$x and $1$1 is at least $7$7."
Think: "At least" means the same as "greater than or equal to". Also "lots of" means there is a multiplication, and "sum" means there is an addition.
Do: $2$2 groups of $x$x is $2x$2x, and the sum of this and $1$1 is $2x+1$2x+1. So altogether we have that "the sum of $2$2 groups of $x$x and $1$1 is at least $7$7" can be written as $2x+1\ge7$2x+1≥7.
We can now solve the inequality for $x$x:
Inequality we set up
Subtract $1$1 from both sides
Divide both sides by $2$2
So the possible values of $x$x are those that are greater than or equal to $3$3.
Consider the following situation:
"$2$2 less than $4$4 groups of $p$p is no more than $18$18".
Construct and solve the inequality described above.
What is the largest value of $p$p that satisfies this condition?
Lachlan is planning on going on vacation. He has saved $\$2118.40$$2118.40, and spends $\$488.30$$488.30 on his airplane ticket.
Let $x$x represent the amount of money Lachlan spends on the rest of his holiday.
Write an inequality to represent the situation, and then solve for $x$x.
What is the most that Lachlan could spend on the rest of his holiday?
At a sport clubhouse the coach wants to rope off a rectangular area that is adjacent to the building. He uses the length of the building as one side of the area, which measures $26$26 meters. He has at most $42$42 meters of rope available to use.
If the width of the roped area is $W$W, form an inequality and solve for the range of possible widths.
Use variables to represent quantities to solve real-world or mathematical problems.
Construct inequalities to solve problems by reasoning about the quantities. o Fluently solve multi-step inequalities with the variable on one side, including those generated by word problems. o Compare an algebraic solution process for equations and an algebraic solution process for inequalities. o Graph the solution set of the inequality and interpret in context.