1.14 Solving inequalities

Operating limits, tolerances, and safety constraints usually describe ranges of acceptable values.

Inequality. A comparison between quantities using \(<\), \(\le\), \(>\), or \(\ge\).

Adding or subtracting the same quantity preserves an inequality. Multiplication or division by a positive value also preserves it. Multiplication or division by a negative value reverses it. In a three-part inequality, apply each operation to all three expressions.

For polynomial and rational inequalities, move all terms to one side, combine fractions, factor, identify zeros and excluded values, divide the number line into intervals, determine the sign on each interval, and check endpoint inclusion. A polynomial can change sign only at a zero. A rational expression can also change sign at an excluded denominator zero. Include a numerator zero only when equality is allowed; never include an undefined value.

Worked Example: A quadratic inequality

Solve \(x^2-12x+35<0\):

\[(x-5)(x-7)<0.\]

The product is negative between its zeros, so \(5<x<7\), or \((5,7)\).

Try It. Solve \((x-2)(x-6)\le0\).
Check Your Work \(2\le x\le6\), or \([2,6]\).

Worked Example: Solving a compound operating limit

Solve \(12\le3q+6<30\). Subtracting 6 from all three expressions and then dividing by 3 gives

\[2\le q<8,\]

or \([2,8)\).

Try It. Solve \(5<2q+1\le13\).
Check Your Work \(2<q\le6\), or \((2,6]\).

For \(c>0\), \(|x|<c\) means \(-c<x<c\), while \(|x|>c\) means \(x<-c\) or \(x>c\). The first statement describes values within \(c\) units of zero, while the second describes values farther than \(c\) units away. More generally, \(|x-a|\le c\) describes a tolerance of \(c\) units around the target \(a\).

1.14.1 Sign charts and tolerances

For \((x-2)(x+3)x(x+5)>0\), the values \(-5,-3,0,2\) divide the number line into five intervals. Test one value in each interval and count negative factors. An even number of negative factors gives a positive product. Because the inequality is strict, none of the four zeros is included.

For a rational inequality, numerator zeros and denominator zeros both divide the sign chart, but denominator zeros are always excluded. For \(|2x-3|<4\), write \(-4<2x-3<4\) and solve all three parts to obtain \(-1/2<x<7/2\). A greater-than absolute-value inequality instead produces two outside intervals.

Use a square bracket only when an endpoint is defined and satisfies a non-strict inequality. Use a parenthesis for strict endpoints, undefined values, and infinity. In applications, intersect the algebraic solution with the physical domain. For example, \(|q-20|\le3\) gives \([17,23]\), but an equipment requirement \(q\ge18\) reduces the usable range to \([18,23]\).

Worked Example: Solving a rational inequality

Solve \(1/(x-3)>0\). The expression is undefined at 3. Its numerator is positive, so the fraction is positive exactly when \(x-3>0\). Therefore, \(x>3\), or \((3,\infty)\).

Try It. Solve \(1/(x+2)<0\).
Check Your Work The denominator must be negative, so \(x<-2\), or \((-\infty,-2)\).

Worked Example: Finding a required pump rate

One pump removes no more than 40 L/min for 8.0 minutes. A second pump runs for 5.0 minutes. To remove at least 480 L,

\[320+5r\ge480,\]

so \(r\ge32\) L/min. The second pump must provide at least 32 L/min when the first operates at its maximum rate.

Try It. If 300 L has already been removed, find the rate needed for a second pump operating 6 minutes to reach at least 480 L.
Check Your Work \(300+6r\ge480\), so \(r\ge30\) L/min.

Worked Example: Applying a chlorine-residual tolerance

A residual must remain within 0.15 mg/L of a 0.80 mg/L target. The condition is

\[|C-0.80|\le0.15,\]

so \(-0.15\le C-0.80\le0.15\) and therefore \(0.65\le C\le0.95\) mg/L.

Try It. A pH value must remain within 0.25 of 7.2. Find the acceptable interval.
Check Your Work \(|p-7.2|\le0.25\) gives \(6.95\le p\le7.45\).

Applied Problem: Combining process and equipment constraints. A chemical pump can operate from 12 through 40 L/h. A treatment calculation shows that the required rate must satisfy \(5r+20\ge145\), while a maximum daily chemical allowance requires \(r<34\) L/h. Find the feasible operating interval.

Check Your Work The treatment requirement gives \(5r\ge125\), so \(r\ge25\). Combining \(r\ge25\), \(r<34\), and the equipment interval \([12,40]\) gives \([25,34)\) L/h.

1.14.2 Practice Problems

  1. Solve \(5x+13\le7\).
    Check Your Work \(x\le-6/5\).
  2. Solve \(-2x+3>11\).
    Check Your Work \(x<-4\).
  3. Solve \((x-2)(x+1)\le0\).
    Check Your Work \([-1,2]\).
  4. Solve \(1/(x-3)>0\).
    Check Your Work \((3,\infty)\).
  5. Solve \(|2x-1|\ge5\).
    Check Your Work \(x\le-2\) or \(x\ge3\).
  6. A pump rate must be at least 28 L/s but less than 45 L/s. Write the interval.
    Check Your Work \([28,45)\) L/s.
  7. A chlorine residual must stay within 0.10 mg/L of 0.75 mg/L. Write and solve an absolute-value inequality.
    Check Your Work \(|C-0.75|\le0.10\), so \(0.65\le C\le0.85\) mg/L.
  8. A tank contains 600 L and fills at 35 L/min. How long must it fill to contain at least 1,300 L?
    Check Your Work \(600+35t\ge1300\), so \(35t\ge700\) and \(t\ge20\) min.